\[
\alpha^2 \text{\struck{$+$}}\, \alpha + 1 = 0 \qquad (\text{racines } \alpha,\ \alpha')
\]

\[
u^2 = u + \alpha
\]

\[
u^4 = u^2 + \alpha^2 = u^2 + \alpha' = u + \alpha + \alpha' = u + 1 = u'
\]

\[
u^5 = u u' = \alpha\, 1
\]

\[
u^3 = u^2 + \alpha u = u + \alpha + \alpha u = \alpha' u + \alpha
\]

\[
\det(u-v) = 0 \quad \text{i.e.} \quad
\underbrace{\det u}_{1} + \underbrace{\det v}_{1} + \operatorname{Tr} u v = 0
\]

\[
\operatorname{Tr} u u' = 0
\]

\[
\operatorname{Tr} u.u = \operatorname{Tr} u^2 = \operatorname{Tr}(u + \alpha.1)
= \operatorname{Tr} u = 1 \neq 0
\]

\[
\det(u+v) = \det u + \det v + \operatorname{Tr} u v = \operatorname{Tr} u v
\qquad \text{\struck{$\det(u-v)$}}
\]

\[
\operatorname{Tr} u v = 0
\]

\[
\operatorname{Tr} u v = 0 \qquad
\operatorname{Tr}(u v') = \operatorname{Tr}(u(v+1))
= \operatorname{Tr} u v + \operatorname{Tr} u = 1 + \operatorname{Tr} u v
\]

\[
\operatorname{Tr} u u' = 0, \qquad \operatorname{Tr} u^2 = 1
\]

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\quad \text{\struck{\ill{}}} \quad
\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \frac{1}{ad-bc}
\]

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix}
= \begin{pmatrix} aa' + bc' & ab' + bd' \\ \cdots & \cdots \end{pmatrix}
\]

\[
\boxed{\begin{aligned}
u u' &= u' u = \det u \\
u + u' &= \operatorname{Tr} u . \mathrm{id}
\end{aligned}}
\qquad 2 \operatorname{Tr} u = 2R
\]

\[
u' = -u \iff \operatorname{Tr} u = 0
\]

\[
u^2 - (\operatorname{Tr} u)\, u + \det u = 0
\]

\[
\operatorname{Tr} u = 1 \qquad \det u = 0 \qquad \det u \overset{?}{=} 1
\qquad u = u'
\]

\[
1 + a' \otimes a, \qquad \langle a, a' \rangle = 1
\]

\[
u \longmapsto -u' \qquad u' u = 0
\]

\[
\text{\struck{$u(x) \wedge y = x \wedge \ldots$}}
\]

\[
u(x) \wedge y = -u'(y) \wedge x
\]

\[
\text{\struck{$\boxed{u(x) \wedge y + x \wedge u'(y) = 0}$}}
\]

\[
\boxed{u(x) \wedge y = x \wedge u'(y)}
\]

\[
E \to \check{E} \simeq E \otimes \omega^{-1}, \qquad
E \times E \to \omega, \qquad
E \simeq \check{E} \otimes \omega, \qquad \check{E}
\]

\[
\lambda e + \mu f, \qquad x \wedge x, \qquad
\text{\struck{$x' \otimes x$}}
\]

\[
15 \cdot 4 = 60, \qquad 4 \cdot 5 = 20
\]

\[
\operatorname{Tr} u = 1 \quad
\begin{array}{|lll r}
\det u = 0 & u^2 - u = 0 & u \text{ un projecteur} & 20 \\
\det u = 1 & u^2 - u + 1 = 0 & u \text{ val. propres } \alpha, \alpha' & 20 \\
\det u = \alpha & u^2 - u + \alpha = 0 & & 12 \\
\det u = \alpha' & u^2 - u + \alpha' = 0 & & 12 \\
 & & & 64
\end{array}
\]

\[
u^2 \text{\struck{$+$}}\, \lambda u + \mu = 0
\]

\[
1,\ u \qquad \alpha.1 \longmapsto \alpha 1 + u \qquad
u \longmapsto \mu.1 + \lambda u
\]

\[
\begin{pmatrix} 0 & -\mu \\ 1 & \lambda \end{pmatrix}
\qquad 1,\ u,\ u'
\]

\[
\operatorname{Tr} u = 1, \quad \operatorname{Tr} v = 1, \qquad
\det u = \det v = \alpha
\]

\[
\operatorname{Tr}(u+v) = 0
\]

\[
\det(u+v) = \underbrace{\det u}_{1} + \underbrace{\det v}_{1}
+ \operatorname{Tr} u v = \operatorname{Tr}(uv)
\]

\[
\operatorname{Tr} u v = \left\{
\begin{array}{l} 0 \\ 1 \\ \alpha \\ \alpha' \end{array}
\right.
\qquad
\begin{array}{|l}
\lambda u + \mu \\
\lambda u' + \mu \\
\lambda^2 \alpha + \mu^2 + \lambda\mu \neq 0
\end{array}
\]

\[
u = u' \Longrightarrow \quad u \quad u' \quad 1
\]

\[
\lambda u + \mu \qquad
\operatorname{Tr}(\lambda u + \mu) = \lambda \operatorname{Tr} u = \lambda = 1
\]

\[
u + \mu \qquad
\det(u+\mu) = \det u + \mu^2 + \mu \operatorname{Tr} u
= \mu^2 + \mu + \alpha = \text{\struck{$\alpha$}}
\]

\[
u + u' = 1 \qquad u + 1 = u'
\]

\[
\text{\struck{$\mu^2 + \alpha\mu \ldots$}} \qquad
\text{\struck{$\alpha^2 = \alpha + \alpha\alpha'$}} \qquad
\alpha'^2 + \alpha\alpha' + 1 = 0
\]

\[
\mu^2 + \mu + \alpha = \alpha, \qquad \mu^2 + \mu = 0
\]

\[
\operatorname{Tr}(uv) = \qquad u \longmapsto u' = u + 1
\qquad u \longmapsto u' \ \text{antipodisme}
\]

\[
uv = \text{\struck{$u+$}}\ \lambda u + \mu, \qquad
u(v - \lambda) = \mu
\]

\[
v - \lambda = \mu u^{-1} = (\mu\alpha')(u+1), \qquad
v = \mu\alpha' u + \ldots
\]

\[
u' u = \alpha. 1, \qquad u^{-1} = \alpha' u'
\]

\[
\operatorname{Tr} u u' = 0, \qquad \operatorname{Tr} u w = 0, \qquad
v = u' + w, \qquad \operatorname{Tr} w = 0, \qquad
\operatorname{Tr} \text{\struck{$u$}}\, w = 0
\]

\[
\operatorname{Tr} \text{\struck{$v$}}\, u v u^{-1} = 0 \text{ ou } 1
\]

\[
\operatorname{Tr} v\, u(v) = \det(v - u(v)) \in \{0, 1\}
\]

\[
k[u] \cap S = \{u, u'\}
\]

\[
u = \begin{pmatrix} 0 & 1 \\ \alpha & 1 \end{pmatrix}
\qquad
u' = \begin{pmatrix} 1 & 1 \\ \alpha & 0 \end{pmatrix}
\qquad
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix}
\]

\[
u^{-1} = \alpha' \begin{pmatrix} 1 & 1 \\ \alpha & 0 \end{pmatrix}
= \begin{pmatrix} \alpha' & \alpha' \\ 1 & 0 \end{pmatrix}
\qquad e_1, \ e_2 \qquad \operatorname{Tr} u
\]

\[
\begin{pmatrix} \lambda & 0 \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 0 & 1 \\ \alpha & 1 \end{pmatrix}
\begin{pmatrix} \lambda^{-1} & 0 \\ 0 & 1 \end{pmatrix}
=
\begin{pmatrix} 0 & \lambda \\ \lambda^{-1}\alpha & 1 \end{pmatrix}
\qquad
v_0 = \begin{pmatrix} 0 & -\lambda \\ \lambda^{-1}\alpha & 1 \end{pmatrix}
\]

\[
v \quad u v u^{-1}, \qquad v' \quad u v' u^{-1}, \qquad
u' u = \det u = \alpha
\]

\[
v u v u^{-1} = v u v (\alpha' u'), \qquad
u(u+1) = \alpha, \qquad u^{-1} = \alpha'(u+1),
\qquad (vu)(vu) + vu
\]

\[
v u = \begin{pmatrix} \lambda\alpha & \lambda \\ \alpha & 1 + \lambda^{-1}\alpha \end{pmatrix}
\qquad
v u^{-1} = \begin{pmatrix} \lambda & 0 \\ 1 + \lambda^{-1} & \lambda^{-1} \end{pmatrix}
\]

\[
\lambda^2 \alpha + \lambda + 1 + \lambda^{-1} + \lambda^{-2}\alpha
= \alpha(\lambda^2 + \lambda^{-2}) + (\lambda + \lambda^{-1}) + 1
\]

\[
\text{\struck{$\lambda\alpha +$}} \quad \lambda + \lambda^{-1} \overset{?}{=} 1
\qquad \boxed{\alpha\sigma^2 + \sigma + 1} \qquad \alpha .
\]

\[
v = \lambda u + \mu \qquad \operatorname{Tr} v = 1 \quad \text{i.e. } \lambda = 1
\]

\[
v = u + \mu, \qquad
\det(v) = \underbrace{\det u}_{1} + \det \mu.1 + \operatorname{Tr} \mu u
= \alpha + \mu^2 + \mu
\]

\[
\mu^2 + \mu = 0 \qquad \mu \in \{0, 1\} \quad \text{i.e.} \quad v \in \{u, u'\}
\]

\[
\mu^2 + \mu + 1 = 0 \qquad \text{i.e.} \quad \mu \in \{\alpha, \alpha'\}
\]

\[
v = u + \alpha, \qquad \underbrace{u + \alpha'}_{u' + \alpha} \in S'
\qquad 0 \quad 1 \quad \alpha \quad \alpha' \ \ldots \quad \alpha^2 = \alpha'
\]

\[
\operatorname{Tr} u v = 0
\]

\[
\operatorname{Tr}(u + \alpha)(v + \alpha) = \operatorname{Tr} u v
+ \text{\struck{\ill{}}}
\qquad uv + \alpha u + \alpha v + \alpha^2
\]

\[
\operatorname{Tr}(u + \alpha, v) = \operatorname{Tr} u v + \alpha
\qquad v \quad u v u^{-1}
\]

\[
\text{\struck{$\operatorname{Tr} v (u v u^{-1}) = \operatorname{Tr}(v u)^2 u^{-2} v'$}}
\qquad u(v)' = u(v')
\]

\[
\begin{array}{c|cccc}
\lambda & 0 & 1 & \alpha & \alpha' \\ \hline
\sigma & & & &
\end{array}
\]

\[
\boxed{\operatorname{Tr} v_0\, u(v_0) = \alpha}
\Longrightarrow
\operatorname{Tr} v\, u(v) = \alpha
\]

\[
\operatorname{Tr} v'\, \underbrace{u(v')}_{u(v)'} = \alpha
\]

\[
\det(u + v) = \operatorname{Tr}(uv) =
\left\{
\begin{array}{ll}
\alpha & \text{si } \{u, v\} \text{ arête} \\
\alpha' & \text{si } \{u, v\} \text{ coarête} \\
1 & \text{si } u = v \\
0 & \text{si } u = u'
\end{array}
\right.
\]

\[
e_3 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad
e_4 = \begin{pmatrix} 1 \\ \lambda \end{pmatrix}
\]

\[
\boxed{u = \begin{pmatrix} \zeta & 0 \\ 0 & -\zeta \end{pmatrix}}
\qquad \operatorname{Tr} u = 0, \quad \det u = -\zeta^2;
\qquad \operatorname{Tr} v = 0, \quad \det v = -\eta^2
\]

\[
v = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad
v(e_1) = a e_1 + c e_2
\]

\[
v(e_3) = (a + b,\ c + d) = (\eta, \eta), \qquad
v(e_4) = (a + \lambda b,\ c + \lambda d) = (-\eta, -\lambda\eta)
\]

\[
\begin{aligned}
a + b &= \eta, & c + d &= \eta, \\
a + \lambda b &= -\eta, & c + \lambda d &= -\lambda\eta
\end{aligned}
\]

\[
b = \eta - a, \quad d = \eta - c
\]

\[
a + \lambda(\eta - a) = -\eta, \quad (1 - \lambda) a = -(1 + \lambda)\eta;
\qquad
c + \lambda(\eta - c) = -\lambda\eta, \quad (1 - \lambda) c = -2\lambda\eta
\]

\[
a = \frac{\lambda + 1}{\lambda - 1}\,\eta, \qquad
c = \frac{2\lambda\eta}{\lambda - 1}
\]

\[
b = \eta\Bigl(1 - \frac{\lambda + 1}{\lambda - 1}\Bigr) = \frac{-2\eta}{\lambda - 1},
\qquad
d = \eta\Bigl(1 - \frac{2\lambda}{\lambda - 1}\Bigr)
= -\frac{\lambda + 1}{\lambda - 1}\,\eta = -a,
\qquad c = -\lambda b
\]

\[
\boxed{v = \begin{pmatrix} a & b \\ -\lambda b & -a \end{pmatrix}}
\qquad
\left\{
\begin{aligned}
a &= \frac{\lambda + 1}{\lambda - 1}\,\eta \\
b &= \frac{-2}{\lambda - 1}\,\eta
\end{aligned}
\right.
\]

\[
\det v = -a^2 + \lambda b^2
= -\eta^2 \underbrace{\Bigl[\Bigl(\frac{\lambda + 1}{\lambda - 1}\Bigr)^2
- \lambda \frac{4}{(\lambda - 1)^2}\Bigr]}_{
\frac{(\lambda + 1)^2 - 4\lambda}{(\lambda - 1)^2} = 1}
\]

\[
\text{\struck{$uv + vu$}} \qquad
u v = \begin{pmatrix} \zeta a & \zeta b \\ \lambda\zeta b & \zeta a \end{pmatrix}
\qquad
v u = \begin{pmatrix} \zeta a & -\zeta b \\ -\lambda\zeta b & \zeta a \end{pmatrix}
\]

\[
uv + vu = \begin{pmatrix} 2\zeta a & 0 \\ 0 & 2\zeta a \end{pmatrix}
= \text{\struck{$2\zeta a \ldots$}}\ 2\zeta a . 1
\qquad 2\zeta a = \operatorname{Tr} u v
\]

\[
\begin{array}{c|c|c|c|c}
(u, v) & v = u & v = u' & \{u, v\} \in A_\alpha & \{u, v\} \in A'_\alpha \\ \hline
\det(u - v) & 0 & 1 & \alpha' & \alpha \\ \hline
\operatorname{Tr} u v & 1 & 0 & \alpha & \alpha'
\end{array}
\]

\[
\begin{aligned}
\operatorname{Tr} v\, u^i(v) &= \alpha & &\text{si } i \equiv 1 \text{ ou } -1 \ (5) \\
\operatorname{Tr} v\, u^i(v) &= \alpha' & &\text{si } i \equiv 2 \text{ ou } -2 \ (5) \\
(&= 1 & &\text{si } i \equiv 0 \ (5))
\end{aligned}
\]

\[
\operatorname{Tr} u v_0 = 1 + \alpha(\lambda + \lambda^{-1}) = 1 + \alpha = \alpha'
\qquad u^5 = \alpha
\]

\[
\operatorname{Tr} u v_0' = \alpha
\qquad
u v = \begin{pmatrix} 1 & 0 \\ \alpha' & \alpha' \end{pmatrix}
\qquad
u^{-1} v = \begin{pmatrix} 0 & 1 \\ 1 & \alpha' \end{pmatrix}
\]

\[
u = \begin{pmatrix} 0 & 1 \\ \alpha & 1 \end{pmatrix}
\qquad
u^{-1} = \begin{pmatrix} \alpha' & \alpha' \\ 1 & 0 \end{pmatrix}
\qquad
v_0 = \begin{pmatrix} 1 & \alpha \\ 1 & 0 \end{pmatrix}
\]

\[
v_1 = u v u^{-1} = \begin{pmatrix} \alpha' & \alpha \\ 1 & \alpha' \end{pmatrix}
\qquad
v_{-1} = u^{-1} v u = \begin{pmatrix} \alpha & 1 \\ \alpha' & \alpha' \end{pmatrix}
\]

\[
u^5 = \alpha, \qquad (\alpha u)^5 = \alpha^6 = 1, \qquad
\det(\alpha u) = \alpha^3 = 1
\]

\[
\frac{12 \cdot 11}{2} = 66 = 6 + 30 + 30, \qquad
\frac{12 \cdot 11 \cdot 10}{6} = 220 = 20 + 20 + 60 + 60 + 6 \cdot 10
\qquad 1 + \alpha = \alpha'
\]

\[
\operatorname{Tr} \text{\struck{$v$}}\, u w u^{-1} = \alpha
\qquad \text{\struck{\ill{}}}\ u' \ \text{\struck{\ill{}}}
\]

\[
\det(u + v + w) = \underbrace{\det u + \det v + \det w}_{3\alpha = \alpha}
+ \underbrace{\varphi(u, v) + \varphi(v, w) + \varphi(w, u)}_{3\alpha' = \alpha'}
= 1
\]

\[
\left\{
\begin{aligned}
\sigma &= \varepsilon_0 \varepsilon_1 \varepsilon_0 \quad (\text{dissymétrique}) \\
\rho_0' &= \sigma \varepsilon_0, \quad \rho_1' = \sigma \varepsilon_1 \\
\rho_0 &= \sigma^{-1} \varepsilon_0 = \varepsilon_0^{-1}\varepsilon_1^{-1}
= (\varepsilon_1 \varepsilon_0)^{-1}, \quad
\rho_1 = \varepsilon_1^{-1}\varepsilon_0^{-1}
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
&\varepsilon_0 \rho_0 = \rho_1 \varepsilon_0 = \rho_0' \varepsilon_0^{-1}
= \rho_1' \varepsilon_1^{-1}, \quad \text{donc } \rho_1 = \varepsilon_0(\rho_0) \\
&\rho_1' = \rho_1^{-2} \\
&\sigma(\varepsilon_0) = \rho_0'(\varepsilon_0)
\end{aligned}
\right.
\]

\[
\sigma(\varepsilon_0) = \varepsilon_1, \quad \sigma(\varepsilon_1) = \varepsilon_0;
\qquad
\sigma(\rho_0) = \rho_1, \quad \sigma(\rho_1) = \rho_0, \quad
\sigma(\rho_0') = \rho_1', \quad \sigma(\rho_1') = \rho_0'
\]

\[
\begin{aligned}
&\rho_0(\varepsilon_0) = \rho_0'(\varepsilon_0) = \varepsilon_1, \qquad
\rho_1(\varepsilon_1) = \rho_1'(\varepsilon_1) = \varepsilon_0, \\
&\varepsilon_0(\rho_0) = \rho_1, \quad \varepsilon_0(\rho_0') = \rho_1', \qquad
\varepsilon_1(\rho_1) = \rho_0, \quad \varepsilon_1(\rho_1') = \rho_0'
\end{aligned}
\]

\[
\rho_0 = \sigma^{-1}\varepsilon_0 = \varepsilon_1 \sigma^{-1}, \quad
\rho_1 = \sigma^{-1}\varepsilon_1 = \varepsilon_0 \sigma^{-1};
\qquad
\rho_0' = \sigma \varepsilon_0 = \varepsilon_1 \sigma, \quad
\rho_1' = \sigma \varepsilon_1 = \varepsilon_0 \sigma
\]

\[
\rho_0^3 = \rho_1^3 = \sigma^{-2} = \omega, \qquad \omega^2 = 1
\]

\[
\Bigl[\ \rho_0^6 = \rho_1^6 = \sigma^{-4} = 1, \qquad
\rho_0'^3 = \rho_1'^3 = 1 \ \Bigr]
\]

\[
\rho_0' = \omega \rho_0, \quad \rho_1' = \omega \rho_1; \qquad
\rho_0' = \rho_0^{-2}, \quad \rho_1' = \rho_1^{-2}
\qquad \text{d'où } \rho_0 = \omega^{-1}\rho_0', \ \rho_1 = \omega^{-1}\rho_1'
\]

\[
\begin{aligned}
\sigma &= \varepsilon_0 \varepsilon_1 \varepsilon_0
\ [= \varepsilon_1 \varepsilon_0 \varepsilon_1] \\
\rho_0 &= \varepsilon_0^{-1}\varepsilon_1^{-1} = (\varepsilon_1 \varepsilon_0)^{-1},
& \rho_1 &= \varepsilon_1^{-1}\varepsilon_0^{-1} = (\varepsilon_0 \varepsilon_1)^{-1} \\
\rho_0^{-1} &= \varepsilon_1 \varepsilon_0, & \rho_1^{-1} &= \varepsilon_0 \varepsilon_1
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_1 &= \sigma(\varepsilon_0) = \sigma \varepsilon_0 \sigma^{-1} \\
\rho_0 &= \sigma^{-1}\varepsilon_0, & \rho_0' &= \sigma \varepsilon_0 \\
\rho_1 &= \varepsilon_0 \sigma^{-1}, & \rho_1' &= \varepsilon_0 \sigma
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_0 &= \sigma \rho_0, \qquad \varepsilon_1 = \rho_0 \sigma, \qquad
\rho_0' = \rho_0^{-2} \\
\rho_1 &= \sigma(\rho_0) = \sigma \rho_0 \sigma^{-1}, \qquad
\rho_1' = \sigma(\rho_0^{-2}) = \sigma \rho_0^{-2} \sigma^{-1}
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_0 &= \sigma^{-1}\rho_0', \qquad \varepsilon_1 = \rho_0' \sigma^{-1}, \qquad
\rho_0 = \sigma^{-2}\rho_0' = \rho_0' \sigma^{-2} \\
\rho_1 &= \sigma^{-1}\rho_0' \sigma^{-1}, \qquad
\rho_1' = \sigma(\rho_0') = \sigma \rho_0' \sigma^{-1}
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_1 &= \rho_0(\varepsilon_0) = \rho_0 \varepsilon_0 \rho_0^{-1}, \qquad
\sigma = \varepsilon_0 \rho_0^{-1}, \qquad \rho_0' = \rho_0^{-2} \\
\rho_1 &= \varepsilon_0(\rho_0) = \varepsilon_0 \rho_0 \varepsilon_0^{-1}, \qquad
\rho_1' = \varepsilon_0(\rho_0^{-2}) = \varepsilon_0 \rho_0^{-2} \varepsilon_0^{-1}
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_1 &= \rho_0'(\varepsilon_0) = \rho_0' \varepsilon_0 \rho_0'^{-1}, \qquad
\sigma = \rho_0' \varepsilon_0^{-1}, \qquad
\rho_0 = \varepsilon_0 \rho_0'^{-1} \varepsilon_0 \\
\rho_1 &= \varepsilon_0^2 \rho_0'^{-1}, \qquad
\rho_1' = \varepsilon_0(\rho_0') = \varepsilon_0 \rho_0' \varepsilon_0^{-1}
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_1 &= \rho_1^{-1}(\varepsilon_0) = \rho_1^{-1} \varepsilon_0 \rho_1, \qquad
\sigma = \rho_1^{-1} \varepsilon_0, \qquad
\rho_0 = \varepsilon_0^{-1}(\rho_1) = \varepsilon_0^{-1} \rho_1 \varepsilon_0 \\
\rho_1' &= \rho_1^{-2}, \qquad
\rho_0' = \varepsilon_0^{-1}(\rho_1^{-2}) = \varepsilon_0^{-1} \rho_1^{-2} \varepsilon_0
\end{aligned}
\]

\[
\begin{aligned}
\varepsilon_1 &= \rho_1'^{-1}(\varepsilon_0) = \rho_1'^{-1} \varepsilon_0 \rho_1', \qquad
\sigma = \varepsilon_0^{-1} \rho_1', \qquad
\rho_1 = \varepsilon_0 \rho_1'^{-1} \varepsilon_0 \\
\rho_0' &= \varepsilon_0^{-1}(\rho_1') = \varepsilon_0^{-1} \rho_1' \varepsilon_0, \qquad
\rho_0 = \rho_1'^{-1} \varepsilon_0^2
\end{aligned}
\]

\[
\begin{aligned}
S\mathcal{T}^{+}_{1,1}
&= \langle \varepsilon_0, \varepsilon_1 \mid
\varepsilon_0 \varepsilon_1 \varepsilon_0 = \varepsilon_1 \varepsilon_0 \varepsilon_1 \rangle \\
&\simeq \langle \rho, \sigma \mid \rho^{-3} = \sigma^2 \rangle \\
&\simeq \langle \rho, \varepsilon_0 \mid
\rho \varepsilon_0 \rho^{-1} \varepsilon_0 \rho = 1 \rangle
\end{aligned}
\]

\[
\lambda^{-1} \quad \lambda^{-\ldots} \quad 1 \quad \lambda^2 \qquad\qquad
\widetilde{\mathbb{Z}}
\]

\[
\varepsilon_0, \ \varepsilon_1, \ \sigma, \ \rho_0, \ \rho_1, \ \rho_0', \ \rho_1', \ \omega
\]

\[
(*)\left\{
\begin{aligned}
&\rho_0' = \sigma \varepsilon_0, \quad \rho_0 = \sigma^{-1}\varepsilon_0 = (\varepsilon_1 \varepsilon_0)^{-1},
\quad \rho_0' = \rho_0^{-2}, \quad \rho_0'^3 = 1, \quad \rho_0^6 = 1 \\
&\rho_1' = \sigma \varepsilon_1, \quad \rho_1 = \sigma^{-1}\varepsilon_1 = (\varepsilon_0 \varepsilon_1)^{-1},
\quad \rho_1' = \rho_1^{-2}, \quad \rho_1'^3 = 1, \quad \rho_1^6 = 1 \\
&\varepsilon_1 = \sigma(\varepsilon_0) = \rho_0(\varepsilon_0) = \rho_0'(\varepsilon_0)
\quad [= \varepsilon_1(\varepsilon_0)] \\
&\varepsilon_0 = \sigma(\varepsilon_1) = \rho_1(\varepsilon_1) = \rho_1'(\varepsilon_1)
\quad [= \varepsilon_0(\varepsilon_1)] \\
&\sigma = \varepsilon_1 \varepsilon_0 \varepsilon_1 = \varepsilon_0 \varepsilon_1 \varepsilon_0
\end{aligned}
\right.
\]

\[
\begin{aligned}
&\rho_0^3 = \rho_1^3 = \sigma^2 = \omega \\
&\omega^2 = 1, \ \omega \text{ commute à tout} \\
&\rho_0' = \omega \rho_0, \quad \rho_0 = \omega \rho_0' \\
&\rho_1' = \omega \rho_1, \quad \rho_1 = \omega \rho_1'
\end{aligned}
\]

\[
\underbrace{\sigma(\varepsilon_0) = \rho_0'(\varepsilon_0)}
= \varepsilon_0 \varepsilon_1 \varepsilon_0 \varepsilon_1^{-1} \varepsilon_0^{-1}
\overset{?}{=} \varepsilon_1
\]

\[
\sigma(\varepsilon_1) = \rho_1'(\varepsilon_1)
= \varepsilon_0 \varepsilon_1 \varepsilon_0 \varepsilon_1 \varepsilon_0^{-1}
\varepsilon_1^{-1} \varepsilon_0^{-1} \overset{?}{=} \varepsilon_0
\]

\[
\rho_0^2 = \varepsilon_1 \varepsilon_0 \varepsilon_1 \varepsilon_0 \overset{?}{=} \rho_0'
= \varepsilon_0 \varepsilon_1 \varepsilon_0 . \varepsilon_0
\quad \text{ssi on a (1)}
\]

\[
\rho_1^2 = \varepsilon_0 \varepsilon_1 \varepsilon_0 \varepsilon_1 \overset{?}{=} \rho_1'
= \varepsilon_0 \varepsilon_1 \varepsilon_0 \varepsilon_1
\quad \text{toujours vrai}
\]

\[
\begin{pmatrix} 0 & \beta \\ -\beta^{-1} & 0 \end{pmatrix}
\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 0 & \beta \\ -\beta^{-1} & 0 \end{pmatrix}
= \begin{pmatrix} -1 & 0 \\ \beta^{-1} & -1 \end{pmatrix}
\qquad \sigma_0(\beta)^{-1} =
\]

\[
\begin{pmatrix} 0 & \beta \\ -\beta^{-1} & -1 \end{pmatrix}.
\]

\[
g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad
g^{-1} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

\[
g \begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix} g^{-1}
= \begin{pmatrix} a & a\beta + b \\ c & c\beta + d \end{pmatrix}
\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
= \begin{pmatrix} 1 - ac\beta & a^2\beta \\ -c^2\beta & 1 + ac\beta \end{pmatrix}
\]

\[
1 - a^2c^2\beta^2 \qquad\qquad ad - bc
\]

\[
\underbrace{\sigma_0(\beta)\, e_0(\beta)\, \sigma_0(\beta)^{-1}}\,
\underbrace{e_0(\beta)\, \sigma_0(\beta)\, e_0(\beta)\, \sigma_0(\beta)^{-1}} = \sigma_0(\beta)
\]

\[
\omega\,\bigl(\sigma_0(\beta)\, e_0(\beta)\bigr)^3 =
\]

\[
\underbrace{\begin{pmatrix} 1 & 0 \\ \gamma & 1 \end{pmatrix}
\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}}_{\begin{pmatrix} 1 & \beta \\ \gamma & \beta\gamma + 1 \end{pmatrix}}
\begin{pmatrix} 1 & 0 \\ \gamma & 1 \end{pmatrix}
= \begin{pmatrix} 1 + \beta\gamma & \beta \\ 2\gamma + \beta\gamma^2 & 1 + \beta\gamma \end{pmatrix},
\qquad 2\gamma + \beta\gamma^2 = \gamma(2 + \beta\gamma)
\]

\[
1 + \beta^2\gamma^2 + 2\beta\gamma - 2\beta\gamma - \beta^2\gamma^2
\]

\[
\rho(\beta) = \begin{pmatrix} 0 & \beta \\ -\beta^{-1} & 0 \end{pmatrix}
\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}
= \begin{pmatrix} 0 & \beta \\ -\beta^{-1} & -1 \end{pmatrix} = u(\beta)\,(
\]

\[
\begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}
\begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}
= \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}
\]

\[
\rho'^3 = \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}
\begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}
= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

\[
\sigma_0(\beta)\bigl(e_0(\beta)\bigr) = e_1(-\beta^{-1})
\]

\[
\begin{pmatrix} 0 & \beta \\ -\beta^{-1} & 0 \end{pmatrix}
\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix} \bigl(
\]

\[
\mathcal{M} = \Bigl(\Gamma,\ Z \subset \mathrm{Aut}(\Gamma),\ \Pi \subset \Pi_Z \subset \mathrm{Bij}(S, \check S)\Bigr),
\qquad \Gamma = (S, \check S, R \subset S \times \check S)
\]

\[
\begin{aligned}
&\chi\{s, t\} \in A, \qquad \chi\{s, \lambda t\} \in A, \qquad \chi\{\lambda s, t\} \in A, \qquad \chi\{s, \lambda^{-1} t\}
\end{aligned}
\]

\[
(s_0, \check s_0),\ (s_0, \lambda\check s_0) \in R; \qquad
\exists\, g \in G_0,\quad g(s_0) = s_0,\ g(\check s_0) = \lambda\check s_0
\]

\[
g^{-1}(s_0) = s_0, \qquad g^{-1}\lambda\, g^{-1}(s_0) =
\]

\[
g s_0 \qquad\qquad \lambda s_0 = s_0,\quad \lambda^{-1} t_0 = t_0
\]

\[
u s_0 = \lambda s_0 \qquad u t_0 = t_0
\]

\[
u \in T.U_0 \qquad u \in U_1
\]

\[
\boxed{TU_0 \cap U_1 = \{1\}}
\]

\[
T \cap U_0 \subset U_1\,?
\]

\[
\underbrace{\mathrm{SL}(M)}_{H} = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Bigm|
\begin{matrix} a, d \in \Lambda \\ c \in L^{-1}L',\ b \in L'^{-1}L \end{matrix},\ ad - bc = 1 \Bigr\}
\]

\[
U_0 = \Bigl\{ \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix} \Bigm| \alpha \in \overbrace{L'^{-1}L}^{L_0} \Bigr\}
\qquad
B_0 = \Bigl\{ \begin{pmatrix} \lambda & \alpha \\ 0 & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \alpha \in L'^{-1}L \Bigr\}
\]

\[
U_1 = \Bigl\{ \begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix} \Bigm| \beta \in \underbrace{L^{-1}L'}_{L_1} \Bigr\}
\qquad
B_1 = \Bigl\{ \begin{pmatrix} \lambda & 0 \\ \beta & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \beta \in L^{-1}L' \Bigr\}
\]

\[
T = \Bigl\{ \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^* \Bigr\}
\qquad
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

\[
\sigma = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \ \text{si } L = L' = \Lambda
\]

\[
N(c) = \begin{pmatrix} 0 & -c^{-1} \\ c & 0 \end{pmatrix} = \sigma . u(c)
\]

\[
g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad
u(\lambda) = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}, \quad
e(\alpha) = \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}, \quad
e'(\beta) = \begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix}
\]

\[
N(c)\,u(\lambda) = N(c\lambda), \qquad u(\lambda)\,N(c) = N(\lambda^{-1}c), \qquad u(\lambda)\,N(c)\,u(\lambda') = N(\lambda^{-1}\lambda' c)
\]

\[
g \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} = \begin{pmatrix} a\lambda & b\lambda^{-1} \\ c\lambda & d\lambda^{-1} \end{pmatrix}, \qquad
\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} g = \begin{pmatrix} \lambda a & \lambda b \\ \lambda^{-1}c & \lambda^{-1}d \end{pmatrix}
\]

\[
\begin{aligned}
g \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} g^{-1}
&= \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \lambda d & -\lambda b \\ -\lambda^{-1}c & \lambda^{-1}a \end{pmatrix} \\
&= \begin{pmatrix} \lambda ad - \lambda^{-1}bc & ab(-\lambda + \lambda^{-1}) \\ cd(\lambda - \lambda^{-1}) & -\lambda bc + \lambda^{-1}ad \end{pmatrix}
\end{aligned}
\]

\[
N = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(M) \Bigm| ab = cd = 0 \Bigr\}
\]

\[
N^+ = T = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(M) \Bigm| b = c = 0 \Bigr\}, \qquad
N^- \,[= \sigma T] = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(M) \Bigm| a = d = 0 \Bigr\}
\]

\[
u(\lambda)\,\sigma = \begin{pmatrix} 0 & -\lambda \\ \lambda^{-1} & 0 \end{pmatrix}, \qquad
\sigma\, u(\lambda) = \begin{pmatrix} 0 & -\lambda^{-1} \\ \lambda & 0 \end{pmatrix}
\]

\[
e(\alpha)\,u(\lambda)\,e'(\beta) = \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}
\underbrace{\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix}}_{\begin{pmatrix} \lambda & 0 \\ \lambda^{-1}\beta & \lambda^{-1} \end{pmatrix}}
= \begin{pmatrix} \lambda + \lambda^{-1}\alpha\beta & \lambda^{-1}\alpha \\ \lambda^{-1}\beta & \lambda^{-1} \end{pmatrix}
= \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

\[
\lambda = d^{-1}, \qquad \alpha = \lambda b = b/d, \qquad \beta = \lambda c = c/d
\]

\[
U_0 \times T \times U_1 \longrightarrow H \quad \text{injectif}
\]

\[
U_0 T U_1 = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| d \in \Lambda^* \Bigr\}
\]

\[
[e(\alpha), e'(\beta)] = \underbrace{e(\alpha)\,e'(\beta)}_{\begin{pmatrix} 1 + \alpha\beta & \alpha \\ \beta & 1 \end{pmatrix}}\, e(-\alpha)\,e'(-\beta)
\]

\[
\begin{pmatrix} 1 + \alpha\beta & \alpha \\ \beta & 1 \end{pmatrix}
\begin{pmatrix} 1 + \alpha\beta & -\alpha \\ -\beta & 1 \end{pmatrix}
= \begin{pmatrix} (1 + \alpha\beta)^2 - \alpha\beta & -\alpha^2\beta \\ \alpha\beta^2 & 1 - \alpha\beta \end{pmatrix}
= \begin{pmatrix} 1 + \alpha\beta + \alpha^2\beta^2 & -\alpha^2\beta \\ \alpha\beta^2 & 1 - \alpha\beta \end{pmatrix}
\]

\[
[e(\alpha), e'(\beta)] = \begin{pmatrix} 3 & -\alpha \\ \alpha^{-1} & 0 \end{pmatrix}
\]

\[
(u_\lambda^{-1} v_{\tilde\lambda}^2)(s_0, t_0) = (\lambda s_0, \lambda t_0)
\]

\[
\lambda^{-1} u_\lambda^{-1} v_{\tilde\lambda}^2\, (s_0, t_0) = (s_0, t_0),
\]

\[
\lambda^{-1} u_\lambda^{-1} v_{\tilde\lambda}^2 \in T' \cap G_0 = \mathfrak{z}_{T'}
\]

\[
\tilde\delta(\lambda^{-1} u_\lambda^{-1} v_{\tilde\lambda}^2) = \tilde\delta(\lambda^{-1})\, \underbrace{\tilde\delta(u_\lambda^{-1})}_{1}\, \underbrace{\tilde\delta(v_{\tilde\lambda})^2}_{\tilde\lambda^2}
\]

\[
\tilde\delta(\lambda) = \tilde\lambda^2 \qquad \forall\, \tilde\lambda \in \tilde Z,\ \lambda = \dot{\tilde\lambda} \in Z \tag{70}
\]

\[
e'(\beta)\, e(\alpha) = \begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix} \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}
= \begin{pmatrix} 1 & \alpha \\ \beta & 1 + \alpha\beta \end{pmatrix}
\]

\[
\boxed{\, e'(\beta)\, e(\alpha) = e\Bigl(\frac{\alpha}{1 + \alpha\beta}\Bigr)\, u\Bigl(\frac{1}{1 + \alpha\beta}\Bigr)\, e'\Bigl(\frac{\beta}{1 + \alpha\beta}\Bigr) \,}
\]

\[
e'(\beta)\, e(\alpha) = \begin{pmatrix} 1 & \alpha \\ \beta & 0 \end{pmatrix} \overset{?}{\in} [\sigma B_1 =]\ N^- U_1
\]

\[
N^- B_1 = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| d = 0 \Bigr\}
= \Bigl\{ \begin{pmatrix} a & b \\ c & 0 \end{pmatrix} \Bigm| a \in \Lambda,\ b \in L'^{-1}L,\ c \in L^{-1}L',\ bc = -1 \Bigr\}
\]

\[
\underbrace{\begin{pmatrix} 0 & b' \\ -b'^{-1} & 0 \end{pmatrix}}_{\in N^-}
\underbrace{\begin{pmatrix} 1 & 0 \\ \beta' & 1 \end{pmatrix}}_{\in U_1}
= \begin{pmatrix} b'\beta' & b' \\ -b'^{-1} & 0 \end{pmatrix}
\]

\[
\begin{pmatrix} a & b \\ c & 0 \end{pmatrix} = \underbrace{\begin{pmatrix} 0 & b \\ c & 0 \end{pmatrix}}_{\in N^-} \underbrace{\begin{pmatrix} 1 & 0 \\ ab^{-1} & 1 \end{pmatrix}}_{\in U_1}
\qquad \text{si } bc = -1
\]

\[
e'(\beta)\, e(\alpha) = \underbrace{\begin{pmatrix} 0 & \alpha \\ \beta & 0 \end{pmatrix}}_{\in N^-} \begin{pmatrix} 1 & 0 \\ \alpha^{-1} = -\beta & 1 \end{pmatrix}
\qquad \text{i.e.}
\]

\[
e'(\beta)\, e(-\beta^{-1}) = \begin{pmatrix} 0 & -\beta^{-1} \\ \beta & 0 \end{pmatrix} \underbrace{\begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix}}_{e'(-\beta)}
\qquad \text{i.e.}
\]

\[
e'(\beta)\, e(-\beta^{-1}) = N(\beta)\, e'(-\beta)
\]

\[
\boxed{\, e'(\beta)\, e(-\beta^{-1})\, e'(\beta) = N(\beta) \,} \qquad \beta \in (L^{-1}L')^* = U_1^*
\]

\[
\lambda + \lambda^{-1}\alpha\beta = d^{-1} + d\,\frac{bc}{d^2} = d^{-1} + \frac{bc}{d} = a
\]

\[
\tilde Z = G/G_0^+ \qquad \text{avec} \qquad G \xrightarrow{\ \tilde\delta\ } \tilde Z \tag{9}
\]

\[
1 \longrightarrow \{\pm 1\} \longrightarrow \tilde Z \longrightarrow Z \longrightarrow 1, \tag{10}
\]

\[
G_0^+ \subset G_0 \subset G
\]

\[
(s_0, t_0) \in \vec A_0 \qquad \text{i.e.} \quad s_0, t_0 \in S,\ \{s_0, t_0\} \in A_0 : \tag{12}
\]

\[
\left\{
\begin{aligned}
B' &= G_{s_0} = \{ g \in G \mid g s_0 = s_0 \} \\
\tilde U &= B' \cap G_0 = \{ g \in G_0 \mid g s_0 = s_0 \} = \{ b' \in B' \mid \delta(b') = 1 \} \\
U &= B' \cap G_0^+ = \{ g \in G_0^+ \mid g s_0 = s_0 \} = \{ b' \in B' \mid \tilde\delta(b') = 1 \} \\
\tilde B'' &= \{ g \in G_0 \mid g s_0 \in Z.s_0 \} \\
B'' &= \tilde B'' \cap G_0^+ \qquad [\mathrm{NB}\ \ \tilde B'' = \tilde U . B'']^{!} \\
B &= B' . B'' \ (= B' . \tilde B'')^{!}
\end{aligned}
\right. \tag{13}
\]

\[
\bigl[e(\alpha)\, u(\lambda)\, \underbrace{e'(\beta)\bigr]\bigl[e(\alpha')}_{e\left(\frac{\alpha'}{1 + \alpha'\beta}\right) u\left(\frac{1}{1 + \alpha'\beta}\right) e'\left(\frac{\beta}{1 + \alpha'\beta}\right)}\, u(\lambda')\, e'(\beta')\bigr] =
\]

\[
\boxed{\, = e\Bigl(\alpha + \lambda^2 \frac{\alpha'}{1 + \alpha'\beta}\Bigr)\, u\Bigl(\frac{\lambda\lambda'}{1 + \alpha'\beta}\Bigr)\, e'\Bigl(\beta' + \lambda'^{-2} \frac{\beta}{1 + \alpha'\beta}\Bigr) \,}
\qquad 1 + \alpha'\beta \in \Lambda^*
\]

\[
e(\alpha)\, u(\lambda)\, \underbrace{e'(\beta)\, e(\alpha')}_{N(\beta)\, e'(-\beta)}\, u(\lambda')\, e'(\beta')
\]

\[
\underbrace{N(\beta)\, u(\lambda')}_{\in N^-}\, \underbrace{e'(\beta' - \lambda'^{-2}\beta)}_{\in U_1}
= e(\alpha)\, u(\lambda)\, \underbrace{N(\beta)\, u(\lambda')}_{N(\beta\lambda^{-1}\lambda')}\, e'(\beta' - \lambda'^{-2}\beta)
\]

\[
e(\alpha)\, N(\underbrace{\beta\lambda^{-1}\lambda'}_{t})\, e'(\underbrace{\beta' - \lambda'^{-2}\beta}_{\gamma})
= N(t) . \underbrace{\underbrace{N(t)^{-1}(e(\alpha))}_{e'(-t^2\alpha)}\, e'(\gamma)}_{e'(-t^2\alpha + \gamma)}
\]

\[
e(\alpha)\, t\, e'(\gamma) = \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}
\underbrace{\begin{pmatrix} 0 & -t^{-1} \\ t & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \gamma & 1 \end{pmatrix}}_{\begin{pmatrix} -t^{-1}\gamma & -t^{-1} \\ t & 0 \end{pmatrix}}
= \begin{pmatrix} -t^{-1}\gamma + \alpha t & -t^{-1} \\ t & 0 \end{pmatrix}
\]

\[
N(t)\bigl(e(\alpha)\bigr) \overset{\mathrm{def}}{=} N(t)\, e(\alpha)\, N(t)^{-1}
= \underbrace{\begin{pmatrix} 0 & -t^{-1} \\ t & 0 \end{pmatrix} \begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix}}_{\begin{pmatrix} 0 & -t^{-1} \\ t & t\alpha \end{pmatrix}}
\begin{pmatrix} 0 & t^{-1} \\ -t & 0 \end{pmatrix}
= \begin{pmatrix} 1 & 0 \\ -t^2\alpha & 1 \end{pmatrix}
\]

\[
\boxed{\, N(t)\, e(\alpha) = e'(-t^2\alpha) \,} \qquad t \in U_1^*,\ \alpha \in U_0
\]

\[
-t^2\alpha + \gamma = -\alpha\beta^2\lambda^{-2}\lambda'^2 + \beta' - \lambda'^{-2}\beta
\]

\[
e(\alpha)\, u(\lambda)\, e'(\beta)\, e(\alpha')\, u(\lambda')\, e'(\beta')
= \boxed{\, N(\beta\lambda^{-1}\lambda')\, e'\bigl(\beta' - \lambda'^{-2}\beta - \alpha\beta^2\lambda'^2\lambda^{-2}\bigr) \,}
\]

\[
\sigma \in B_G \qquad \text{\struck{$\sigma =$}}\ \sigma t \sigma t \qquad \sigma = t . u . t \qquad u\, g\, u^{-1} = g^{-1}
\]

\[
\underbrace{\begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix} \begin{pmatrix} 1 & -\beta^{-1} \\ 0 & 1 \end{pmatrix}}_{\begin{pmatrix} 1 & -\beta^{-1} \\ \beta & 0 \end{pmatrix}}
\begin{pmatrix} 1 & 0 \\ \beta & 1 \end{pmatrix}
\overset{?}{=} \begin{pmatrix} 0 & -\beta^{-1} \\ \beta & 0 \end{pmatrix} \qquad \text{OK}
\]

\[
T_G = T_0 . T_H, \qquad \underbrace{\mathrm{Im}(T_H \to \mathrm{Aut}(H))}_{\subset\, \mathrm{Im}(T_0 \to \mathrm{Aut}(H))},
\]

\[
\ill{} = \ill{}
\]

\[
e_M \in \bigl(\textstyle\bigwedge^2 M\bigr)^* . \tag{1}
\]

\[
\bigotimes_{i \in \varepsilon_T} L_i \simeq \bigl(\textstyle\bigwedge^2 M\bigr)_{\Lambda_{\pm 1}}^{\varepsilon_T} \simeq \Lambda_{\Lambda_{\pm 1}}^{\varepsilon_T} \tag{2}
\]

\[
\textstyle\bigwedge^2 M \xrightarrow{\ \sim\ } \Lambda \qquad e_M \longmapsto 1 \tag{3}
\]

\[
T \xrightarrow{\ \chi_i\ } \mathbb{G}_m \qquad i \in \varepsilon_T \tag{4}
\]

\[
\prod_{i \in \varepsilon_T} \chi_i = 1 \tag{5}
\]

\[
M = L'_0 \oplus L'_1 , \tag{6}
\]

\[
T \xrightarrow{\ \sim\ } \mathbb{G}_m, \tag{7}
\]

\[
H \overset{\mathrm{déf}}{=} \underbrace{\mathrm{SL}(M)}_{= \mathrm{Aut}(M, e_M)} \simeq
\Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Bigm|
\begin{matrix} a, d \in \Lambda,\ b \in \mathrm{Hom}(L'_1, L'_0) \simeq L_1'^{-1} L'_0 \\ c \in \mathrm{Hom}(L'_0, L'_1) \simeq L_0'^{-1} L'_1,\ ad - bc = 1 \end{matrix} \Bigr\} \tag{8}
\]

\[
bc = cb \in \Lambda \tag{9}
\]

\[
\Lambda^* \xrightarrow{\ \sim\ } T(\Lambda) = \Bigl\{ \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^* \Bigr\} \tag{10}
\]

\[
L_0 = L_1'^{-1} L'_0, \qquad L_1 = L_0'^{-1} L'_1, \tag{11}
\]

\[
e_0 : L_0 \longrightarrow H, \quad \beta \longmapsto \begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix}, \qquad
e_1 : L_1 \longrightarrow H, \quad \gamma \longmapsto \begin{pmatrix} 1 & 0 \\ \gamma & 1 \end{pmatrix} \tag{12}
\]

\[
\begin{aligned}
U_0 &= e_0(L_0) = \Bigl\{ \begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix} \Bigm| \beta \in L_0 \Bigr\} \\
U_1 &= e_1(L_1) = \Bigl\{ \begin{pmatrix} 1 & 0 \\ \gamma & 1 \end{pmatrix} \Bigm| \gamma \in L_1 \Bigr\} .
\end{aligned} \tag{13}
\]

\[
\left\{
\begin{aligned}
u(\lambda)\, g &= \begin{pmatrix} \lambda a & \lambda b \\ \lambda^{-1}c & \lambda^{-1}d \end{pmatrix} \qquad
g\, u(\lambda) = \begin{pmatrix} \lambda a & \lambda^{-1}b \\ \lambda c & \lambda^{-1}d \end{pmatrix} \\
u(\lambda)\, g\, u(\lambda)^{-1} &= \begin{pmatrix} a & \lambda^2 b \\ \lambda^{-2}c & d \end{pmatrix}
\end{aligned}
\right. \tag{14}
\]

\[
\left\{
\begin{aligned}
u(\lambda)\bigl(e_0(\beta)\bigr) &= e_0(\lambda^2\beta) \\
u(\lambda)\bigl(e_1(\gamma)\bigr) &= e_0(\lambda^{-2}\gamma)
\end{aligned}
\right. \tag{15}
\]

\[
T \cap U_0 = T \cap U_1 = \{1\}, \tag{16}
\]

\[
\left\{
\begin{aligned}
B_0 = T . U_0 &= \{ e_0(\beta)\, u(\lambda) \mid \beta \in \Lambda,\ \lambda \in \Lambda^* \}
= \Bigl\{ \begin{pmatrix} \lambda & \lambda^{-1}\beta \\ 0 & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \beta \in \Lambda \Bigr\} \\
&\qquad \Bigl( = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| c = 0 \Bigr\} \Bigr) \\
B_1 = T . U_1 &= \{ u(\lambda)\, e_1(\gamma) \mid \gamma \in \Lambda,\ \lambda \in \Lambda^* \}
= \Bigl\{ \begin{pmatrix} \lambda & 0 \\ \lambda^{-1}\gamma & \lambda^{-1} \end{pmatrix} \Bigm| \lambda \in \Lambda^*,\ \gamma \in \Lambda \Bigr\} \\
&= \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| b = 0 \Bigr\}
\end{aligned}
\right. \tag{17}
\]

\[
B_0 \cap B_1 = T = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| b = c = 0 \Bigr\} . \tag{18}
\]

\[
\left\{
\begin{aligned}
&\sigma_0 : L_0^* \longrightarrow H \\
&\sigma_0(\beta) = \begin{pmatrix} 0 & \beta \\ -\beta^{-1} & 0 \end{pmatrix}
\end{aligned}
\right. \tag{19}
\]

\[
\left\{
\begin{aligned}
\sigma_0(\beta)^2 &= \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = u(-1) \overset{\mathrm{déf}}{=} \omega \qquad (\text{élément \emph{central} de } H) \\
\sigma_0(\beta)\bigl(u(\lambda)\bigr) &= u(\lambda^{-1}) \qquad \text{donc } \sigma(\beta) \text{ normalise } T \\
\sigma_0(\lambda\beta) &= u(\lambda)\, \sigma_0(\beta) = \sigma_0(\beta)\, u(\lambda^{-1}) ,
\end{aligned}
\right. \tag{20}
\]

\[
N^- = \sigma_0(L_0^*) = \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| a = 0,\ d = 0 \Bigr\}, \tag{21}
\]

\[
L_0^* \xrightarrow{\ \sim\ } N^- \tag{22}
\]

\[
\sigma_1 : L_1 \xrightarrow{\ \sim\ } N^- \qquad \sigma_1(\gamma) = \begin{pmatrix} 0 & -\gamma^{-1} \\ \gamma & 0 \end{pmatrix} \tag{23}
\]

\[
\left\{
\begin{aligned}
\sigma_1(\gamma) &= \sigma_0(-\gamma^{-1}) \\
\sigma_1(\lambda\gamma) &= u(\lambda^{-1})\, \sigma_1(\gamma) = \sigma_1(\gamma)\, u(\lambda)
\end{aligned}
\right. \tag{24}
\]

\[
\boxed{\, \sigma_0(\beta)\bigl(e_0(\beta)\bigr) = e_1(-\beta^{-1}) \,} \tag{25}
\]

\[
\forall\, \sigma \in N^-, \quad \text{on a} \quad \sigma(U_0) = U_1, \quad \text{d'où} \quad \sigma(U_1) = U_0 \tag{26}
\]

\[
g = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Longrightarrow g^{-1} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \tag{27}
\]

\[
g\, e_0(\beta)\, g^{-1} = \begin{pmatrix} 1 - ac\beta & a^2\beta \\ -c^2\beta & 1 + ac\beta \end{pmatrix} \tag{28}
\]

\[
a^2\beta = 0, \quad ac\beta = 0 \qquad \forall\, \beta \in L_0
\]

\[
a^2 = 0, \quad ac = 0
\]

\[
\left\{
\begin{aligned}
\mathrm{Transp}_H(U_0, U_1) &= \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| a = 0 \Bigr\}, \quad \text{de même} \\
\mathrm{Transp}_H(U_1, U_0) &= \Bigl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in H \Bigm| d = 0 \Bigr\} .
\end{aligned}
\right. \tag{29}
\]

\[
g\, e_0(\beta)\, g^{-1} = \begin{pmatrix} 1 & 0 \\ -c^2\beta & 1 \end{pmatrix} \tag{28'}
\]

\[
N^- = \mathrm{Transp}(U_1, U_0) \cap \mathrm{Transp}(U_0, U_1) \tag{30}
\]

\[
\left\{
\begin{aligned}
&\beta\gamma = -1 \qquad \text{i.e. } \gamma = -\beta^{-1}, \text{ i.e. } \beta = -\gamma^{-1} \\
&\text{i.e. } e_1(\gamma) = \underbrace{\sigma_0(\beta)\bigl(e_0(\beta)\bigr)}_{e_1(-\beta^{-1})} ,
\end{aligned}
\right. \tag{31}
\]

\[
e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) = \sigma_0(\beta) \qquad (\text{pour } \beta\gamma = -1) \tag{32}
\]

\[
\underbrace{\sigma_0(\beta)\, e_0(\beta)}\, \sigma_0(\beta)^{-1}\, e_0(\beta)\, \underbrace{\sigma_0(\beta)\, e_0(\beta)}\, \sigma_0(\beta)^{-1} = \sigma_0(\beta)
\]

\[
\bigl(\sigma_0(\beta)\, e_0(\beta)\bigr)^3 = 1
\]

\[
\rho(\beta)^3 = 1, \qquad \text{où } \rho(\beta) = \sigma_0(\beta)\, e_0(\beta), \tag{33}
\]

\[
\rho(\beta) = \begin{pmatrix} 0 & \beta \\ -\beta^{-1} & -1 \end{pmatrix} = u(\beta)\, \rho' \qquad \rho' = \rho(1) = \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix} \tag{34}
\]

\[
e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) = \begin{pmatrix} 1 + \beta\gamma & \beta \\ \gamma(2 + \beta\gamma) & 1 + \beta\gamma \end{pmatrix} \tag{35}
\]

\[
\text{si } \beta \in L_0,\ \gamma \in L_1, \text{ les conditions suivantes sont équivalentes} \tag{37}
\]

\[
\left|
\begin{aligned}
&\text{a) } \beta\gamma = -1 \\
&\text{b) } e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) \in \mathrm{Transp}(U_0, U_1) \\
&\text{c) } e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) \in \mathrm{Transp}(U_1, U_0) \\
&\text{d) } e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) \in N^- \\
&\text{e) } e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) = \sigma_0(\beta)
\end{aligned}
\right.
\]

\[
u_0 u_1 u_0 u_1^{-1} u_0^{-1} u_1^{-1} = 1 \qquad \text{i.e. } u_0 u_1 u_0 = u_1 u_0 u_1 \tag{38}
\]

\[
e_0(\beta)\, e_1(\gamma)\, e_0(\beta) = e_1(\gamma)\, e_0(\beta)\, e_1(\gamma) \quad \text{ssi} \quad \beta\gamma = -1 \qquad (\beta \in L_0^*,\ \gamma \in L_1) . \tag{39}
\]

\[
u_0 \longmapsto u_0^\vee : U_0^* \longrightarrow U_1^*
\]

\[
(u_0^\lambda)^\vee = (u_0^\vee)^{\lambda^{-1}}, \qquad u_0 \in U_0^*,\ \lambda \in \Lambda^* \tag{40}
\]

\[
\mathrm{int}(g) | U_0 : U_0 \xrightarrow{\ \sim\ } U_1
\]

\[
N' = T \sqcup N^- \subset H \tag{41}
\]

\[
V \longrightarrow V_J \overset{\mathrm{déf}}{=} \prod_{i \in J} V_i
\]

\[
0 \longrightarrow V \xrightarrow{\ \varphi\ } \prod V_i \longrightarrow D \longrightarrow 0, \qquad \varphi = (\varphi_i), \quad D = \prod V_i / \mathrm{Im}\, V \tag{1}
\]

\[
u_i : V_i \xrightarrow{\ \mathrm{inj}_i\ } \prod V_i \longrightarrow D
\]

\[
0 \longrightarrow K(D, I) \longrightarrow D^I \xrightarrow{\ \varepsilon_{D,I}\ } D \longrightarrow 0
\]

\[
K(D, I) \overset{\mathrm{déf}}{=} \operatorname{Ker} \varepsilon_{D,I} = \Bigl\{ (x_i)_{i \in I} \in D^I \Bigm| \textstyle\sum x_i = 0 \Bigr\}
\]

\[
G_j,\ G_h \qquad G_j = G_{\bar\rho(i)},\quad G_h = G_{\rho(i)}
\]

\[
V_i \mathbin{\wedge_{G_k}} V_j \mathbin{\wedge_{G_i}} V_k \simeq \mathbb{1}_{G_i} \qquad \text{isom\supplied{orphisme} de bitorseurs}
\]

\[
x_i \wedge x_j \wedge x_h = \varepsilon
\]

\[
\mathfrak{S}_I \ (\simeq \mathfrak{S}_3) \qquad \text{\struck{$\mathrm{Aut}\, G_i \times \mathrm{Aut}\, G_i$}} \qquad \text{et}
\]

\[
\underbrace{\prod_{i \in I}{}_{\Gamma}\, \mathrm{Aut}(G_i)}_{\simeq\, \Gamma \text{ si les } G_i \text{ \uncertain{commutatifs}}}, \qquad \text{où} \quad
\Gamma \simeq \underbrace{\mathrm{Aut}\,\mathrm{Ext}(G_i)}_{\text{indépendant de } i \text{ à isom. près}}
\]

\[
\prod G_i, \quad \text{opérant sur } (V_i) \text{ via} \quad g_j x_i g_h^{-1},\quad g_h x_j g_i^{-1},\quad g_i x_h g_j^{-1}
\]

\[
V \xrightarrow{\ \varphi_i\ } V_i \qquad i \in I
\]

\[
I = \{ i_1, i_2, i_3, i_4 \}
\]

\[
u_{(i_1, i_2), (i_3, i_4)} : V_{i_1} \xrightarrow{\ \sim\ } V_{i_2} \tag{1}
\]

\[
\text{(2)}\qquad u_{(i_2,i_1)(i_3,i_4)} = u_{(i_1,i_2)(i_3,i_4)}^{-1}
\]

\[
\text{(3)}\qquad u_{(i_3,i_2)(i_1,i_4)}\,u_{(i_2,i_1)(i_3,i_4)} = \mathrm{id}
\]

\[
\text{(4)}\qquad \rho_{ijkl} \in \mathrm{Aut}\,V_i \qquad (\{i,j,k,l\} = I).
\]

\[
u_{ijkl} : V_i \xrightarrow{\ \sim\ } V_j, \qquad u_{ijlk} : V_i \xrightarrow{\ \sim\ } V_j,
\]

\[
\text{(4)}\qquad \rho_{ijkl} = u_{ijlk}^{-1}\,u_{ijkl}
\]

\[
\text{(5)}\qquad \rho_{ijlk} = (\rho_{ijkl})^{-1}
\]

\[
\text{(6)}\qquad \rho_{jikl} = \text{\struck{$\ldots$}}
\begin{cases}
u_{ijkl}(\rho_{ijkl}^{-1})\ \text{\struck{$\ldots$}}\\
u_{ijlk}(\rho_{ijkl}^{-1})
\end{cases}
\qquad \rho_{jikl} = u_{jilk}^{-1}\,u_{jikl}
\]

\[
\rho_{ji,lk} = u_{ijkl}(\rho_{ijkl}) = u_{ijlk}(\rho_{ijkl})
\]

\[
\rho_{i'_1 i'_2\, i''_1 i''_2}
\]

\[
\rho_{ijkl} \quad\text{et}\quad \rho_{ikjl}
\]

\[
\rho_{ijkl} + \rho_{ikjl} = 1
\]

\[
\text{(7)}\qquad \boxed{\rho_{ikjl} = 1 - \rho_{ijkl}}
\]

\[
\text{(8)}\qquad
\begin{cases}
\rho_{jikl} = \rho_{ijlk} = \rho_{ijkl}^{-1}\\
\rho_{ikjl} = 1 - \rho_{ijkl}
\end{cases}
\qquad (\Rightarrow\ \rho_{jilk} = \rho_{ijkl})
\]

\[
\sigma_{12}(\lambda) = \sigma_{34}(\lambda) = \lambda^{-1}, \qquad \sigma_{23}(\lambda) = 1-\lambda
\]

\[
\underbrace{\mathrm{Rep}(I)}_{\text{torseur sous }\mathfrak S_4} \longrightarrow U(k)
\]

\[
\text{(9)}\qquad \rho_{(i,j)(k,l)} = \rho_{(k,l)(i,j)}
\]

\[
\underbrace{\mathrm{Rep}\,\mathrm{Cor}(I)}_{\text{torseur sous }\mathfrak S_3} \longrightarrow U(k)
\]

\[
\{I, E, \{E_i\}_{i\in I}, (\varphi_i)_{i\in I}\}
\]

\[
\varphi_i : E \to E_i \qquad i\in I
\]

\[
\text{(1)}\qquad E \xrightarrow{\ \varphi_J\ } E_J = \prod_{i\in J} E_i
\]

\[
\text{(2)}\qquad S = \coprod_{i\in I} E_i
\]

\[
\text{(3)}\qquad R\subset S\times E
\]

\[
(s, x)\in R \iff s = \varphi_i(x) \qquad (s = (i\in I,\ s\in E_i),\ x\in E).
\]

\[
E \longrightarrow \mathfrak P(S), \qquad x \longmapsto R(x)\subset S ;
\]

\[
u \longrightarrow I \qquad (\text{i.e. } u\cap E_i \text{ de card} \leq 1 \text{ pour tt } i)
\]

\[
\varphi_i : E \longrightarrow E_i
\]

\[
\begin{aligned}
\varepsilon^S_{\mathcal I',\mathcal I} &: \widetilde S_{\mathcal I} \xrightarrow{\ \sim\ } \widetilde S_{\mathcal I'}\\
\varepsilon^A_{\mathcal I',\mathcal I} &: \widetilde A_{\mathcal I} \xrightarrow{\ \sim\ } \widetilde A_{\mathcal I'}\\
\varepsilon^F_{\mathcal I',\mathcal I} &: \widetilde F_{\mathcal I} \xrightarrow{\ \sim\ } \widetilde F_{\mathcal I'}
\end{aligned}
\]

\[
\varepsilon^S_{\mathcal I',\mathcal I}(s) : \widetilde S_{\mathcal I}(s) \longrightarrow \widetilde S_{\mathcal I'}(s)
\]

\[
u \longmapsto 2u .
\]

\[
\varepsilon^S_{\mathcal I,\mathcal I'}(s)\,\varepsilon^S_{\mathcal I',\mathcal I}(s) = u\longmapsto 4u = -u = a_{\mathcal I'}u
\]

\[
\text{(1)}\qquad
\begin{cases}
\varepsilon_{\mathcal I',\mathcal I}\,\varepsilon_{\mathcal I,\mathcal I'} = a_{\mathcal I'}\\
\varepsilon_{\mathcal I,\mathcal I'}\,\varepsilon_{\mathcal I',\mathcal I} = a_{\mathcal I}
\end{cases}
\qquad\text{et de même.}
\]

\[
\alpha_{\mathcal I',\mathcal I} : t_{\mathcal I} \simeq t'_{\mathcal I'}
\]

\[
\varepsilon^F_{\mathcal I',\mathcal I} : \underset{\textstyle\widetilde A_{\mathcal I,t}}{\omega(t_{\mathcal I})} \xrightarrow{\ \sim\ } \underset{\textstyle\widetilde A_{\mathcal I',t'}}{\omega(t'_{\mathcal I'})}
\]

\[
\begin{cases}
\varepsilon^F_{\mathcal I',\mathcal I}\circ\varepsilon^F_{\mathcal I,\mathcal I'} = \mathrm{id}_{F_{\mathcal I'}}\\
\varepsilon^F_{\mathcal I,\mathcal I'}\circ\varepsilon^F_{\mathcal I',\mathcal I} = \mathrm{id}_{F_{\mathcal I}}
\end{cases}
\qquad\text{et de même.}
\]

\[
\begin{aligned}
\widetilde A_{\mathcal I,a} &\simeq a\wedge\alpha_{\mathcal I} \quad (\simeq\omega(Q_{\mathcal I})) \quad\text{et de même}\\
\widetilde A_{\mathcal I',a} &\simeq a\wedge\alpha_{\mathcal I'}
\end{aligned}
\]

\[
\widetilde A_{\mathcal I,a}\wedge\widetilde A_{\mathcal I',a} \simeq \alpha_{\mathcal I}\wedge\alpha_{\mathcal I'} \simeq a
\]

\[
\varepsilon^{\vec A}_{\mathcal I',\mathcal I} : \vec{\widetilde A}_{\mathcal I} \xrightarrow{\ \sim\ } \vec{\widetilde A}_{\mathcal I'}
\]

\[
\begin{cases}
\varepsilon^{\vec A}_{\mathcal I',\mathcal I}\,\varepsilon^{\vec A}_{\mathcal I,\mathcal I'} = \mathrm{id}_{\vec{\widetilde A}_{\mathcal I'}}\\
\varepsilon^{\vec A}_{\mathcal I,\mathcal I'}\,\varepsilon^{\vec A}_{\mathcal I',\mathcal I} = \mathrm{id}_{\vec{\widetilde A}_{\mathcal I}}
\end{cases}
\qquad\text{et de même.}
\]

\[
\begin{cases}
\vec I' \xrightarrow[\sim]{\ \alpha_{I'}\ } \mathrm{compl.}(\vec I)\\
\vec I \xrightarrow[\sim]{\ \alpha_I\ } \mathrm{compl.}(\vec I')
\end{cases}
\]

\[
\mathrm{compl}(\alpha_{I'})\,\alpha_I : I \longrightarrow \underbrace{\mathrm{compl}\,\mathrm{compl}(I)}_{\textstyle -I}
\]

\[
1 \longrightarrow \underset{\substack{\simeq\,\mathrm{Aut}(I')\\ \simeq\,\mathrm{Aut}^+(I)\times\pm1\\ \simeq\,\mathrm{Aut}^+(I')\times\pm1}}{\mathrm{Aut}(I)} \longrightarrow \mathrm{Aut}(I,I') \longrightarrow \mathbb Z/2\mathbb Z \longrightarrow 1
\]

\[
\mathrm{Aut}(I,I')\,/\,\underbrace{\mathrm{Aut}^{0+}(I,I')}_{\substack{\text{automorphismes qui respectent } I,\ I'\\ \text{et l'orientation de } I \text{, i.e. de } I'}}
\]

\[
\begin{gathered}
0 \longrightarrow \mathrm{Aut}(\vec I) \longrightarrow \mathrm{Aut}(\underbrace{\{\vec I,\vec I^\circ\}}_{\text{bicos. or.}}) \longrightarrow \mathbb Z/2\mathbb Z \longrightarrow 0\\
0 \longrightarrow \mathrm{Aut}(\vec I) \longrightarrow \mathrm{Aut}(\{I,I^\circ\}) \longrightarrow \mathbb Z/4\mathbb Z \longrightarrow 0
\end{gathered}
\]

\[
\text{bi-icosaèdres (droits) non orienté} \longmapsto \text{bi-icosaèdre orienté}
\]

\[
1 \longrightarrow \underbrace{\mathrm{Sl}(2,\mathbb F_5)/\pm1}_{\mathrm{Aut}(\vec I)} \longrightarrow \mathrm{Gl}(2,\mathbb F_5)/\pm1 \longrightarrow \underset{\textstyle\mathbb Z/4\mathbb Z\ !!!}{\mathbb F_5^*} \longrightarrow 1
\]

\[
G = \mathrm{Aut}(I)\times\underset{\textstyle\simeq\mathbb Z/4\mathbb Z}{\mathbb F_5^*} .
\]

\[
a\otimes a \in \mathrm{End}(E) = M_2(\mathbb Z/n\mathbb Z)
\]

\[
  \lambda\, C^*_i = C^*_i \quad\text{ssi}\quad \lambda \in k^{*2} \qquad (\text{pour } \lambda\in k^*) .
\]

\[
  \rho_a = \sigma \text{ d'ordre } 4, \qquad \rho_f \text{ d'ordre } 3,
\]

\[
  \left\{
  \begin{array}{ll}
    \det\sigma = 1 , & \det\rho_f = 1 \\
    \operatorname{Tr}\sigma = 0 , & \operatorname{Tr}\rho_f = -1
  \end{array}
  \right.
\]

\[
  \sigma . \rho_f = \rho_s^{-1} \qquad \rho_s \text{ unipotent i.e. } \det\rho_s = 1,\ \operatorname{Tr}\rho_s = 2 .
\]

\[
  M \simeq \operatorname{End}(E) .
\]

\[
  L' = L\otimes \underline{\omega}^{-1} \qquad\text{donc}
\]

\[
  D \simeq L^{\otimes 2}\otimes\underline{\omega}^{-1}
  \quad\text{i.e.}\quad
  \underline{\omega} \xrightarrow[\sim]{\varphi_L} \underline{\operatorname{Hom}}(L^{\otimes 2}, D_L)
\]

\[
  \varphi_L^e : \struck{\ill{}} \; L^{\otimes 2} \xrightarrow{\ \sim\ } D_L
\]

\[
  \varphi_L^{\lambda e} = \lambda\, \varphi_L^{e} .
\]

\[
  D^*(e) = \{\, \varphi_L^e(x\otimes x) \mid x\in L^* \,\} \qquad\text{pour } D = D_L
\]

\[
  D^*(\lambda e) = \lambda D^*(e)
\]

\[
  D^*/k^{*2} \simeq \underline{\omega}^*/k^{*2} .
\]

\[
  C^*(e) = \bigcup_{D} D^*(e)
\]

\[
  \left\{
  \begin{array}{l}
    C^*(\lambda e) = \lambda C^*(e) \\
    \lambda^2 C^*(e) = C^*(e)
  \end{array}
  \right.
\]

\[
  C^* = \coprod_{e\in\underline{\omega}^*/k^{*2}} C^*(e)
\]

\[
  \varphi^e : E^* \longrightarrow C^*(e), \qquad x\longmapsto \varphi^e(x\otimes x)
\]

\[
  \left\{
  \begin{array}{l}
    \varphi^e(\lambda x) = \lambda^2 \varphi^e(x) \\
    \varphi^{\lambda e}(x) = \lambda\, \varphi^e(x)
  \end{array}
  \right.
\]

\[
  \Theta \longmapsto u(\Theta) = u\,\Theta\, u^{-1} .
\]

\[
  \varphi^e : E^* \xrightarrow{\ \sim\ } C^*(e) \subset C^*
\]

\[
  \boxed{\ \Theta(\varphi^e(x)) = \det\Theta\,\bigl(\varphi^e(\Theta(x))\bigr)\ }
\]

\[
  \Bigl(\, \Theta(x\overset{e}{\otimes} x) = \Theta(x)\overset{\Theta(e)}{\otimes}\Theta(x)
  = \Theta(x)\overset{\det\Theta.e}{\otimes}\Theta(x)
  = \det\Theta\,\bigl(\Theta(x)\overset{e}{\otimes}\Theta(x)\bigr) \Bigr)
\]

\[
  \text{\struck{$(a\overset{e}{\otimes}b)\circ(c\overset{e}{\otimes}d) \simeq c\overset{e}{\otimes}b\,(d\wedge a/e)$}} ,
\]

\[
  \text{\struck{$u\circ w = \bigl(y\overset{e}{\otimes}(-x\wedge y/e)\bigr)\, z\overset{e}{\otimes}z$}}
\]

\[
  \text{\struck{$= -z\otimes x\,(z\wedge y/e) = z\overset{e}{\otimes}x$}}
\]

\[
  \rho x = y, \quad \rho y = z, \quad \rho z = x
\]

\[
  \det\rho = 1, \quad \operatorname{Tr}\rho = 1, \quad\text{donc}\quad \rho^{-1} = 1-\rho = \rho' \quad (\text{conj. de } \rho)
\]

\[
  \tau = x\overset{e}{\otimes}x + y\overset{e}{\otimes}y + z\overset{e}{\otimes}z
\]

\[
  \tau = a\rho + b
\]

\[
  \operatorname{Tr}\tau = a + 2b \quad\text{donc}\quad a = -2b, \text{ donc}
\]

\[
  \tau = \struck{a\rho+2b}\; b(-2\rho+1)
\]

\[
  \det\tau = b^2\det(1-2\rho) \qquad\text{où } \det(1-2\rho) = 3
\]

\[
  \tau = \pm(1-2\rho)
\]

\[
  \tau = 1-2\rho
\]

\[
  \text{\struck{$u+v+w = 1-2\rho$}}
\]

\[
  \rho\in M^* \text{ unique}
  \left\{
  \begin{array}{l}
    \det\rho = 1 \\
    \rho^3 = 1
  \end{array}
  \right.
  \qquad
  \begin{array}{l}
    \rho \text{ induit une permutation} \\
    \text{circulaire entre les } u, v, w
  \end{array}
\]

\[
  \boxed{\ u+v+w = 1-2\rho\ }
\]

\[
  \{\pm 1\} \xrightarrow[\sim]{\text{iso}} \mu_2(k) .
\]

\[
  \psi^e : E^*/\pm 1 \xrightarrow{\ \sim\ } C^*(e)
\]

\[
  T_\lambda(\bar y) = \overline{y+\lambda x}
\]

\[
  \underbrace{x\overset{e}{\otimes}x}_{=\varphi^e(x)} \circ \underbrace{y\overset{e}{\otimes}y}_{=\varphi^e(y)}
  = y\overset{e}{\otimes}x\,\bigl((x\wedge y)/e\bigr)
\]

\[
  \operatorname{Tr}(u\circ v) = -\bigl((x\wedge y)/e\bigr)^2
\]

\[
  x\wedge y = \pm e \iff \boxed{\ \operatorname{Tr} uv = -1\ }
\]

\[
  \operatorname{Tr} uv = \operatorname{Tr} vw = \operatorname{Tr} wu = -1
\]

\[
  x+y+z = 0 \qquad x\wedge y = y\wedge z = z\wedge x = e
\]

\[
  u = x\overset{e}{\otimes}x, \quad v = y\overset{e}{\otimes}y, \quad w = z\overset{e}{\otimes}z
\]

\[
  \delta(xyzt) = \delta(xy(zt)) = Q\bigl(\delta(x),\delta(y),\delta(zt),\delta(xy),\delta(yzt),\delta(ztx)\bigr)
\]

\[
  \text{\struck{$= \delta(x(yz),t)$}}
\]

\[
  = \delta(x(yz)t) = Q\bigl(\delta(x),\delta(yz),\delta(t),\delta(xyz),\delta(yzt),\delta(tx)\bigr)
\]

\[
  = \delta((xy)zt) = Q\bigl(\delta(xy),\delta(z),\delta(t),\delta(xyz),\delta(zt),\delta(txy)\bigr)
\]

\[
  \delta(x),\delta(y),\delta(z),\delta(t),\ \delta(xy)\,\delta(xz)\,\delta(xt)\,\delta(yz)\,\delta(yt)\,\delta(zt)
\]

\[
  \boxed{\ \delta(\tau h) \overset{?}{=} \delta(\tau)\ }
\]

\[
  \delta(\tau h) - \delta(\tau) = \text{expression polynomiale en}
\]

\[
  \underbrace{\delta(\{\tau^{n}(h), h\})}_{\text{commutateurs}} \text{ et } \delta(\tau)\,\delta(\tau h),\ \delta(h)
\]

\[
  \boxed{\ \delta(\tau.\underbrace{\tau(h)h^{-1}}_{\{\tau,h\}}) \overset{?}{=} \delta(\tau)\ }
  \qquad (\text{si } \tau \text{ normalise } H \text{ commut., } h\in H)
\]

\[
  \left[\ x*y = \bar x\,\bar y\,\bar x^{-1}\,\bar y^{-1} \qquad
  \delta(\bar x) = \frac{b^2-4c}{c} \ \right]
  \qquad c = \det x,\ b = \operatorname{Tr} x
\]

\[
  1*1 = 1
\]

\[
  (x*y)(y*x) = 1
\]

\[
  xs*y = x*y \quad\text{si } s, y \text{ commutent}
\]

\[
  [\, x*yt = x*y \quad\text{si } t, x \text{ ——}\,]
\]

\[
  [\, x*y = 1 \quad\text{si } x, y \text{ commutent}\,]
\]

\[
  (x*y)^2 - P(\delta x,\delta y,\delta(xy))\, x*y + 1 = 0
  \qquad P\in\mathbb{Z}[X,Y,Z] \text{ bien déterminé}
\]

\[
  [\,\delta(z), x*y\,] = [\,\delta(z),\delta(z')\,] = 0
\]

\[
  \left\{
  \begin{array}{l}
    \delta(xy) = \delta(yx) \qquad\text{i.e. } \delta(g(x)) = \delta(x) \qquad (x,y,g\in G) \\
    \delta(x) = \delta(x^{-1}) \\
    \delta(xy) + \underbrace{\delta(xy^{-1})}_{=\delta(x^{-1}y)} = (\delta(x)+2)(\delta(y)+2) - 4
  \end{array}
  \right.
\]

\[
  = \delta(x)\delta(y) + 2(\delta(x)+\delta(y))
\]

\[
  \Bigl[\ \boxed{\delta(1) = 0} \Bigr]
\]

\[
  \delta(x_i)\ (1\leq i\leq 4), \qquad \delta(x_i x_j)\ (1\leq i<j\leq 4)
\]

\[
  \boxed{\ \delta(xyz) \overset{?}{=} Q\bigl(\delta(x),\delta(y),\delta(z),\delta(xy),\delta(yz),\delta(zx)\bigr)\ }
\]

\[
  \delta(x^2) = (\delta(x)+2)^2 - 4 = \delta(x)^2 + 4\delta(x)
  \qquad [\text{cas particulier de } \delta(xy)+\delta(xy^{-1}) = \ldots]
\]

\[
  \delta(x^n) = P_n(\delta(x)) \qquad n\in\mathbb{Z} \qquad P_n = P_{-n}
\]

\[
  \delta(x^n y^m) = P_{n,m}(\delta(x),\delta(y),\delta(xy))
\]

\[
  \delta(\underbrace{M(x_1,\ldots,x_n)}_{\substack{\text{monôme non commutatif}\\ \text{à exposants entiers}\,\in\mathbb{Z}}})
  = P_M\bigl((\delta(x_i)),(\delta(x_i x_j))\bigr)
\]

\[
  (1)\quad
  \left\{
  \begin{array}{lll}
    \operatorname{Tr} : A\longrightarrow k & k\text{-linéaire} & \\
    \det : A\longrightarrow k & \text{multiplicatif} & \text{, quadratique}
  \end{array}
  \right.
\]

\[
  (2)\quad u^2 - \operatorname{Tr}(u).u + \det(u) = 1
\]

\[
  (3)\quad \det(u+v) = \det u + \det v + (\operatorname{Tr}u\operatorname{Tr}v - \operatorname{Tr}(uv))
\]

\[
  (u,v)\longmapsto \Phi(u,v) \overset{\text{déf}}{=} \operatorname{Tr}u\operatorname{Tr}v - \operatorname{Tr}(uv)
\]

\[
  (4)\quad (\operatorname{Tr}u)^2 - \operatorname{Tr}(u^2) = 2\det u ,
\]

\[
  (2')\quad u^2 = \operatorname{Tr}(u)u - \det(u)
\]

\[
(5)\qquad \begin{cases} \operatorname{Tr} 1 = 2 \\ \det 1 = 1 \end{cases}
\]

\[
(6)\qquad uv + vu - (\operatorname{Tr} u)v - \operatorname{Tr}(v)u + \bigl(\operatorname{Tr} u \operatorname{Tr} v - \operatorname{Tr}(uv)\bigr)1 = 0
\]

\[
(6')\qquad uv + vu = \operatorname{Tr}(v)u + (\operatorname{Tr} u)v + \bigl(\operatorname{Tr}(uv) - \operatorname{Tr} u \operatorname{Tr} v\bigr)1 .
\]

\[
u^2,\ v^2,\ uvuv,\ uv,\ u^2v,\ vu,\ vuv,\ uvu,\ uv^2
\]

\[
\begin{aligned}
u^2 &= \operatorname{Tr}(u).u - \det(u).1 && \text{(2) pour } u\\
v^2 &= \operatorname{Tr}(v).v - \det(v).1 && \text{(2) pour } v\\
uv &= 1.uv && \text{tautol.}\\
vu &= -1.uv + \operatorname{Tr}(v)u + \operatorname{Tr}(u)v + \bigl(\operatorname{Tr}(uv) - \operatorname{Tr} u \operatorname{Tr} v\bigr)1 && \text{(relation (6'))}
\end{aligned}
\]

\[
u^2,\ uv,\ u^2v,\ vu,\ v^2,\ vuv = \struck{\ill{}}\,(vu)v .
\]

\[
\text{\struck{$vu = \ill{} + \operatorname{Tr}(v).u +$}}
\]

\[
(6'')\qquad vu = \bigl(\operatorname{Tr}(uv) - \operatorname{Tr} u \operatorname{Tr} v\bigr)1 + \operatorname{Tr}(v)u + \operatorname{Tr}(u)v - uv .
\]

\[
(7)\qquad \begin{cases} b = \operatorname{Tr} u,\ c = \det u \\ b' = \operatorname{Tr} v,\ c' = \det v \\ t = \operatorname{Tr}(uv) \end{cases}
\]

\[
(8)\qquad M(b, c;\, b', c';\, t) .
\]

\[
(9)\qquad (b, c, b', c', t) \in k^5 .
\]

\[
\mathbf{Z}[B, C, B', C', T]
\]

\[
M = k[u] \oplus k[v]
\]

\[
(10)\qquad \det(u_1 + v_1) = \underbrace{\det u_1}_{N_{A_1/k}(u_1)} + \underbrace{\det v_1}_{N_{A_2/k}(v_1)} + \operatorname{Tr} u_1 \operatorname{Tr} v_1 - \operatorname{Tr}(u_1v_1)
\]

\[
\delta \overset{\text{déf}}{=} b^2 - 4c \text{ inv.}, \qquad \delta' \overset{\text{déf}}{=} b'^2 - 4c' \text{ inv.}
\]

\[
\operatorname{Tr}(u\Theta(v)), \qquad \Theta(v) \overset{\text{déf}}{=} \Theta v \Theta^{-1} .
\]

\[
\check{\mathbb{V}}(A) \longrightarrow \mathbb{E}^1_k \;(= \operatorname{Spec} k[T])
\]

\[
\rho = f(b, c, b', c';\, t) = \frac{P(b, c, b', c', t)}{Q(\ldots)}
\]

\[
P - 2Q \neq 2 \qquad \text{\uncertain{partout}}
\]

\[
\Bigl(\rho = \lambda + \text{\struck{$\ill{}$}}\,\lambda^{-1} \qquad \text{d'où} \quad \lambda^2 - \rho\lambda + 1 = 0
\]

\[
\rho = 2\,\frac{t^2 + 4cc'}{t^2 - 4cc'}
\qquad\Bigl|\qquad
\begin{gathered}\text{NB } t^2 - 4cc' = \delta(uv)\\ \text{si } b = b' = 0, \text{ discriminant de } uv\end{gathered}
\]

\[
\frac{\rho}{2} - 1 = 8\,\frac{cc'}{t^2 - 4cc'} .
\]

\[
(11)\qquad \rho = 2\,\frac{(2t - bb')^2 + \delta(u)\delta(v)}{(2t - bb')^2 - \delta(u)\delta(v)}
\]

\[
= \frac{2t^2 - 2bb't + \bigl(b^2b'^2 - 2(b^2c' + b'^2c) + 8cc'\bigr)}{t^2 - bb't + (b^2c' + b'^2c - 4cc')}
\]

\[
\rho = \frac{(bb')^2}{t^2 - bb't + (b^2c' + b'^2c)} \; .]
\]

\[
(**)\qquad \rho_0 = \rho - 2 = \frac{b^2b'^2 - 4(b^2c' + b'^2c) + 16cc'}{t^2 - bb't + (b^2c' + b'^2c - 4cc')} = \frac{A(b, b', c, c')}{B(t; b, b', c, c')}
\]

\[
\rho\, B(t; b, b', c, c') - A(b, b', c, c') = 0
\]

\[
\lambda^2 - \rho\lambda + 1 = 0 \qquad \text{\struck{\ill{}}}
\]

\[
\operatorname{Tr} u = b, \quad \det u = c
\]

\[
\operatorname{Tr} v = b', \quad \det v = c' .
\]

\[
\tau = \operatorname{Tr} uv
\]

\[
t^2 - bb't + \Bigl(b^2c' + b'^2c - 4cc' - \frac{\delta\delta'}{\rho_0}\Bigr)
\]

\[
t + t' = bb' \qquad (\tau = t') .
\]

\[
(11')\qquad
\begin{cases}
\rho_0 = \rho - 2 = \dfrac{\delta\,\delta'}{t^2 - bb't + (\delta c' + \delta'c + 4cc')}\\[2ex]
\phantom{\rho_0 = \rho - 2} = \dfrac{(b^2 - 4c)(b'^2 - 4c')}{\underbrace{t^2 - bb't + b^2c' + b'^2c - 4cc'}_{Q\,=\,Q(t;\, b, b', c, c')}}
\end{cases}
\]

\[
(b^2 - 4c)(b'^2 - 4c')(t^2 - bb't + b^2c' + b'^2c - 4cc')
\]

\[
u' = (\operatorname{Tr} u)1 - u = b.1 - u
\qquad
\begin{cases} \operatorname{Tr} u' = \operatorname{Tr} u \\ \det u' = \det u \end{cases}
\]

\[
t' = \operatorname{Tr} u'v = \operatorname{Tr}\bigl((b - u)v\bigr) = b\operatorname{Tr}(v) - \operatorname{Tr}(uv) = bb' - \tau
\]

\[
(12_1)\qquad (b^2 - 4c)(b'^2 - 4c') \in k^*
\]

\[
(12_2)\qquad \underbrace{t^2 - bb't + b^2c' + b'^2c - 4cc'}_{Q(t;\, b, c, b', c')} \in k^* \;?
\]

\[
\text{\struck{$\rho_0(t^2 - bb't$}} \quad \rho_0\, Q = \delta\delta'
\]

\[
\Delta(Q) = (bb')^2 - 4(b^2c' + b'^2c - 4cc')
\]

\[
(13)\qquad \Delta(Q) = (b^2 - 4c)(b'^2 - 4c') = \delta\delta'
\]

\[
\text{\struck{$t \neq bb' - t$ \ i.e. \ $2t \neq bb'$}}
\]

\[
(14)\qquad \delta\delta' \in k^* \Longrightarrow 2t \neq bb' \ \text{\struck{\uncertain{pour tout} $t$}} \qquad \text{i.e. } 2t - bb' \in k^*
\]

\[
Q(t;\, b, c, b', c') = 0
\]

\[
\begin{array}{c|c|cccc}
 & 1 & \underline{u} & \underline{v} & \underline{uv}\\ \hline
\underline{1} & \underline{1} & \underline{u} & \underline{v} & \underline{uv}\\
\underline{u} & \underline{u} & -c + b\underline{u} & \underline{uv} & -c\underline{v} + b\underline{uv}\\
\underline{v} & \underline{v} & (t - bb')\underline{1} + b'\underline{u} + b\underline{v} - \underline{uv} & -c'\underline{1} + b'\underline{v} & -bc'\underline{1} + c'\underline{u} + t\underline{v}\\
\underline{uv} & \underline{uv} & -b'c\underline{1} + t\underline{u} + c\underline{v} & -c'\underline{u} + b'\underline{uv} & t\underline{uv} - cc'
\end{array}
\]

\[
\begin{pmatrix}
2 & b & b' & t\\
b & b^2 - 2c & t & bt - b'c\\
b' & t & b'^2 - 2c' & b't - bc'\\
t & bt - b'c & b't - bc' & t^2 - cc'
\end{pmatrix}
\]

\[
\begin{pmatrix}
4 & 2b & 2b'\\
2b & b^2 & bb'\\
2b' & bb' & b'^2
\end{pmatrix}
\]

\[
\text{discriminant de $\operatorname{Tr}$ par rapport à la base $1, u, v$ de $k1 \oplus ku \oplus kv$} = -2\,Q(t;\, b, c, b', c')
\]

\[
\det\Theta = \operatorname{Tr}\Theta = 0
\]

\[
\delta = \delta' = 0 \qquad \text{i.e.} \quad b^2 - 4c = b'^2 - 4c' = 0
\]

\[
\alpha^2 = 0, \qquad \beta(2\alpha + b) = 0
\]

\[
\operatorname{Tr}\Theta = \det\Theta = 0
\]

\[
M^0 = \{x \in M \mid \operatorname{Tr} x = 0\}
\]

\[
\operatorname{Tr} u = \operatorname{Tr} v = 0, \qquad \det u = \det v = 0 .
\]

\[
Q(t, b, b', c, c') = t^2 .
\]

\[
\operatorname{Tr}(ux) = 0 \iff x \text{ normalise } ku
\]

\[
x \in B_u \qquad \text{(sous-alg. de Borel normalisatrice de $u$)}
\]

\[
k \oplus ku \oplus kuv = B_u = B_v
\]

\[
M[b, c, b', c';\, t] = M(0, 0, 0, 0, t) \text{ est Azumaya,}
\]

\[
Q(0, 0, 0, 0, t) = t^2 \in k^*
\]

\[
u = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \quad \text{et} \quad v = t\begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix}
\qquad
\operatorname{Tr} uv = t \quad \text{OK.}
\]

\[
b' = c' = 0
\]

\[
Q(b, c, b', c', t) = t^2
\]

\[
\Delta(b, c, b', c', t) = -\bigl(t^2 - bb't + b^2c' + b'^2c - 4cc'\bigr)
\]

\[
\text{\struck{$b = 1,\ c = 0$}}, \quad b' = 0, \quad c' = 0
\]

\[
Q = t^2, \qquad \Delta = -Q = -t^2
\]

\[
uv = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}
\]

\[
v = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \qquad u = \begin{pmatrix} \alpha & \beta\\ \gamma & \delta \end{pmatrix} \quad \text{« général »,}
\]

\[
uv = \begin{pmatrix} 0 & \alpha\\ 0 & \gamma \end{pmatrix} \qquad \operatorname{Tr} uv = \gamma
\]

\[
\delta(u) = (\alpha + \delta)^2 - 4(\alpha\delta - \beta\gamma)
\]

\[
\varphi(u,v)=\operatorname{Tr}(uv) \quad\text{et}\quad \psi(u,v)=\varphi_{\det}(u,v)=\operatorname{Tr}u\operatorname{Tr}v-\operatorname{Tr}(uv),
\]

\[
A\simeq k.1\oplus A^{\natural}, \qquad B=k.1\oplus V,\quad V\subset A^{\natural},
\]

\[
A^{\natural}\overset{\mathrm{def}}{=}\{u\in A \mid \operatorname{Tr}u=0\} = 1^{\perp}
\]

\[
B^{\perp(\varphi)}=B^{\perp(\psi)} .
\]

\[
\varphi(1,u)=\psi(1,u)=\operatorname{Tr}u
\]

\[
\psi(u,\xi)=-\varphi(u,\xi)
\]

\[
\mathbb{P}(\dot{A}/k)
\]

\[
\Gamma(\underline{\mathcal{O}}_{P}(2))\simeq \operatorname{Sym}^{2}_{k}(A/k)\simeq \text{espace des formes quadratiques sur } (A/k)^{\vee}\simeq A^{\natural} .
\]

\[
\mu:\underbrace{\textstyle\bigwedge^{2}B/k}_{=\ \det B/k}\longrightarrow A/B
\]

\[
\det B\simeq \underbrace{\det k}_{k}\otimes\det B/k\simeq \det B/k
\]

\[
\mu:\det B\longrightarrow A/B
\]

\[
\mu\in A/B^{\otimes 2}\simeq D^{-\otimes 2}.
\]

\[
B \text{ sous-algèbre} \iff \mu=0 .
\]

\[
\mu=\pm\delta
\]

\[
(\det B)^{\otimes 2}\otimes (A/B)^{\otimes 2}\simeq \underline{1}
\]

\[
(\det B)^{\otimes -2}\simeq (A/B)^{\otimes 2}.
\]

\[
\delta_{\det B}\in (A/B)^{\otimes 2}\simeq D^{\otimes -2}
\]

\[
\begin{aligned}
B\times B &\longrightarrow A/B\\
(u,v)&\longmapsto uv \bmod B
\end{aligned}
\]

\[
u^{2}\equiv 0 \bmod B
\]

\[
\textstyle\bigwedge^{2}B\xrightarrow{\ \mu\ }A/B .
\]

\[
\mu(u\wedge v)=uv \bmod B .
\]

\[
B_{D}=D^{\perp}
\]

\[
\delta_{B}\in(\det B)^{\otimes -2}\Bigl[\underset{\text{iso can}}{\simeq} A/B^{\otimes 2}\Bigr]
\]

\[
\det(A)\simeq k
\]

\[
\det B\otimes A/B\simeq\det A\ (\simeq k)
\]

\[
\operatorname{disc}_{\varphi_{B}}=\operatorname{disc}_{\varphi_{D}}=2\operatorname{disc}'_{\det}(B)
\]

\[
D\subset D^{\perp}=B .
\]

\[
2d(x)=\varphi_{d}(x,x)=0 .
\]

\[
\det_{B}=\det|B ,
\]

\[
\operatorname{disc}'(\det_{B}) \quad\text{tel que}\quad 2\operatorname{disc}'(\det_{B})=\operatorname{disc}\psi_{B}
\]

\[
\operatorname{disc}\psi_{B}=\operatorname{disc}\varphi_{B}=2\underbrace{\operatorname{disc}'(\det_{B})}_{\overset{\mathrm{def}}{=}\ \delta(B)}
\]

\[
H\cap H'=k
\]

\[
H\oplus H'\overset{\mathrm{déf}}{=}B \quad\text{sur la fibre sous-fibré de rang 3}
\]

\[
V\subset A^{\natural},\quad V'\subset A^{\natural}
\]

\[
V+V'=A^{\natural}
\]

\[
V\cap V'\overset{\mathrm{def}}{=}D
\]

\[
D_{1},D_{2}\subset V\cap \underset{\substack{\parallel\\ \text{con.\ isotrope}}}{C},\qquad D_{1}',D_{2}'\subset V'\cap C
\]

\[
\lambda=\lambda_{(D_{1},D_{2})}(D_{1}',D_{2}')\in k \qquad \lambda \text{ partout } \neq 0,1 \text{ i.e. } \lambda \text{ et } 1-\lambda \text{ inv.}
\]

\[
D_{1}\longleftrightarrow D_{2} \qquad\text{ou}\qquad D_{1}'\longleftrightarrow D_{2}'
\]

\[
\rho=\lambda+\lambda^{-1}
\]

\[
\lambda^{2}-\lambda\rho+1=0 .
\]

\[
\rho\neq 2 \quad\text{sur chaque fibre,}
\]

\[
\rho_{0}=\rho-2 \quad\text{est }\neq 0\text{ sur chaque fibre i.e.\ est inv.}
\]

\[
\Theta(H)=\overline{H} \qquad \Theta(H')=\overline{H'}
\]

\[
\overline{\rho_{0}}=\rho_{0}
\]

\[
x\longmapsto x'=\operatorname{Tr}(x).1-x
\]

\[
\underline{\mathrm{Aut}}(A)\simeq \check{V}(A)^{*}/\mathbb{G}_{m} \quad\text{est trivial}
\]

\[
P\xrightarrow{\ \sim\ }\operatorname{Isom}(\varepsilon_{H},\varepsilon_{\overline{H}})\ (=\varepsilon_{H}\wedge\varepsilon_{\overline{H}})
\]

\[
\text{\struck{$P\times\varepsilon_{H}\longrightarrow\varepsilon_{\overline{H}}$}}
\]

\[
P\xrightarrow{\ \sim\ }\varepsilon_{H'}\wedge\varepsilon_{\overline{H}'} \quad ]
\]

\[
H'=\overline{H}' \quad\text{correspondant : la bisection}
\]

\[
\begin{aligned}
Z^{2}-bZ+c&=0 \qquad b^{2}-4c\neq 0\\
Z^{2}-\overline{b}Z+\overline{c}&=0 \qquad \overline{b}^{2}-4\overline{c}\neq 0
\end{aligned}
\]

\[
\rho=\frac{b^{2}}{c}-2,\qquad \overline{\rho}=\frac{\overline{b}^{2}}{\overline{c}}-2
\]

\[
\frac{b^{2}}{c}=\frac{\overline{b}^{2}}{\overline{c}},
\]

\[
1,u,v \qquad\text{base de } B=H+H'
\]

\[
\begin{cases}
b=\operatorname{Tr}u,\ c=\det u & \delta_{H}=b^{2}-4c\\
b'=\operatorname{Tr}v,\ c'=\det v & \delta_{H'}=b'^{2}-4c'\\
t=\operatorname{Tr}uv=\operatorname{Tr}vu
\end{cases}
\]

\[
\boxed{\rho_{0}=\frac{P}{Q}}
\]

\[
\begin{aligned}
P&=P(b,c,b',c',t)=(b^{2}-4c)(b'^{2}-4c')=\delta_{H}\delta_{H'}\\
Q&=Q(b,c,b',c',t)=t^{2}-bb't+(b^{2}c'+b'^{2}c-4cc')
\end{aligned}
\]

\[
Q=\tfrac{1}{2}\delta_{\varphi|B} \text{ pour la base } 1,u,v
=\tfrac{1}{2}\det\begin{pmatrix}\operatorname{Tr}1&\operatorname{Tr}u&\operatorname{Tr}v\\ \operatorname{Tr}u&\operatorname{Tr}u^{2}&\operatorname{Tr}uv\\ \operatorname{Tr}v&\operatorname{Tr}vu&\operatorname{Tr}v^{2}\end{pmatrix}
\]

\[
=\text{discr.\ divisé de la forme } \det_{|B} \text{ p.r.\ à la base } 1,u,v
\]

\[
\begin{pmatrix}2&b&b'\\ b&b^{2}-2c&t\\ b'&t&b'^{2}-2c'\end{pmatrix}\ )
\]

\[
\det|H
\]

\[
\det H^{\otimes(-2)}\simeq (H/k)^{\otimes -2}
\]

\[
\det\begin{pmatrix}\operatorname{Tr}1&\operatorname{Tr}u\\ \operatorname{Tr}u&\operatorname{Tr}u^{2}\end{pmatrix}=b^{2}-4c
\]

\[
\delta_{H}=(b^{2}-4c)(1\wedge u)^{\otimes -2}
\]

\[
B=H+H'
\]

\[
B/k\simeq H/k+H'/k
\]

\[
\det B\simeq \underbrace{\det k}_{k}\otimes\det B/k\simeq H/k\otimes H'/k
\]

\[
(\det B)^{\otimes -2}\simeq (H/k)^{\otimes -2}\otimes(H'/k)^{\otimes -2}
\]

\[
\begin{cases}
\delta_{H}\delta_{H'}\in(\det B)^{\otimes -2}\simeq D^{\otimes -2}\\
\delta_{B}\in(\det B)^{\otimes -2}\simeq D^{\otimes -2}
\end{cases}
\]

\[
\boxed{\rho_{0}=\frac{\delta_{H}\delta_{H'}}{\delta_{B}}}
\]

\[
-\delta_{B=H+H'}=Q=t^{2}-bb't+(b^{2}c'+b'^{2}c-4cc')
\]

\[
\lambda^{2}-(\rho_{0}+2)\lambda+1=0
\]

\[ \tau = \operatorname{Tr} uv , \]

\[ \text{\struck{$b^2-4c$}}\ \frac{\delta\delta'}{Q(t)} = \frac{\delta\delta'}{Q(\tau)} \qquad (=\rho_0) \]

\[ Q(t) = Q(\tau) \qquad (=\sigma) \]

\[ R(T) = T^2 - bb'T + r \]

\[ t+\tau = bb' \qquad\text{donc}\qquad \tau = bb'-t . \]

\[ u' = \operatorname{Tr} u.1 - u = b - u \]

\[ \operatorname{Tr} u' = \operatorname{Tr} u = b , \qquad \det u' = \det u = c \]

\[ \operatorname{Tr} u'v = \operatorname{Tr}(b-u)v = b\underbrace{\operatorname{Tr} v}_{b'} - \underbrace{\operatorname{Tr} uv}_{t} = bb' - t = \tau \]

\[ M(b,c,b',c',t) \]

\[ \underline{1},\ \underline{u},\ \underline{v},\ \underline{uv} \]

\[ H' = \text{\struck{\ill{}}}\ k[V]/V^2 \]

\[ Q(t) = t^2 - t = (t-1)t \]

\[ Q(t) = t^2 \]

\[ \operatorname{Tr} uv = t \]

\[ \operatorname{Tr} uv = t \]

\[ \operatorname{Tr} uv = t \]

\[ u = x'\otimes x , \qquad v = y'\otimes y \qquad\qquad x,y\in V,\ x',y'\in V^{\vee} \]

\[ u\circ v = \langle\text{\struck{$y$}},\text{\struck{$x'$}}\rangle\, \text{\struck{$y'$}}\otimes x \]

\[ t = \operatorname{Tr} uv = \langle x,y'\rangle\langle y,x'\rangle \]

\[ y_1y_1' + y_2y_2' = 1 \]

\[ t = y_1y_1' \]

\[ \begin{cases} y_1y_1' = t \\ y_2y_2' = 1-t \end{cases} \]

\[ y_1' = t/y_1 , \qquad y_2' = \frac{1-t}{y_2} \]

\[ y_1y_1' = 0 , \qquad y_2y_2' = 1 \]

\[ y_1y_1' + y_2y_2' = 0 , \qquad t = y_1y_1' \]

\[ \begin{cases} y_1y_1' = t \\ y_2y_2' = -t \end{cases} \]

\[ \begin{cases} y_1y_1' + y_2y_2' = 0 \\ t = y_1'y_2 \end{cases} \]

\[ \Delta_H : H\to k , \qquad T_H : H\to k , \qquad \varepsilon_H : k\to H \]

\[ \Delta_{H'} : H'\to k , \qquad T_{H'} : H'\to k , \qquad \varepsilon_{H'} : k\to H' \]

\[ \varphi_{\Delta_H}(u,v) = \operatorname{Tr}(u)\operatorname{Tr}(v) - \underbrace{\operatorname{Tr}(uv)}_{\psi_H(u,v)} \]

\[ \varphi_{\Delta_H} = \varphi_H \]

\[ \operatorname{Tr}_H(uv) = T_H(u)T_H(v) - \varphi_{\Delta_H}(u,v) \]

\[ \varphi_H(u,v) + \psi_H(u,v) = T_H(u)T_H(v) \]

\[ B = H \amalg_k H' = H\oplus H' / k(1_H \ominus 1_{H'}) \]

\[ H\cap H' = k1_B , \qquad H+H' = B \]

\[ 1_B = 1_H = 1_{H'} \quad \text{\uncertain{i.e. image commune}} \]

\[ P = H\otimes_k H' \]

\[ i(u) = u\otimes 1 , \qquad j(v) = 1\otimes v \]

\[ i(\lambda 1_H) = j(\lambda 1_{H'}) = \lambda\, \underbrace{1\otimes 1}_{\overset{\text{déf}}{=}\,1_P} \]

\[ P \xrightarrow{\ f\ } A \]

\[ f(u\otimes v) = i(u).j(v) \]

\[ i = f\circ i_0 , \qquad j = f\circ j_0 \]

\[ \varphi_P(U,V) = \operatorname{Tr}(f(U).f(V)) \]

\[ (1) \qquad \Delta_P(u\otimes v) = \Delta_H(u)\,\Delta_{H'}(v) \]

\[ L = H/k \simeq B/H' , \qquad L' = H'/k \simeq B/H \]

\[ \text{\struck{$\lambda : L\otimes L' \to k$}} \]

\[ (L\otimes L')^{\vee} \simeq \text{Esp. des formes bil. \add{sym.} sur $B$ nulles sur $H'$ \struck{\ill{}} et $H$ \struck{\ill{}}} \]

\[ \bigl((x,y),(\xi,\eta)\bigr) \longmapsto \lambda(x\otimes\eta) + \lambda(\xi\otimes y) \qquad x,\xi\in L,\ y,\eta\in L' \]

\[ (L\otimes L')^{\vee} \simeq \text{formes quadratiques sur $B$ qui sont nulles sur $H$, $H'$} \]

\[ \underbrace{\gamma(\psi)}_{\overset{\text{déf}}{=}\ \varphi_B} + \psi_B = \bigl((u,v)\longmapsto T_B(u)\,T_B(v)\bigr) \]

\[ \delta(\Delta_P) = \{ u \longmapsto \varphi_{\Delta_P}(u,1_P) \} \]

\[ \Delta_H(1_H) = \Delta_{H'}(1_{H'}) = 1 . \]

\[ P \xrightarrow{\ q\ } Q \]

\[ q(x\otimes y) = 0 \qquad \forall x,y \in P . \]

\[ H\to L , \qquad H'\to L' \]

\[ P \overset{\text{déf}}{=} H\otimes H' \longrightarrow L\otimes L' \quad \text{\uncertain{épi}} \]

\[ 0 \to B \to P \to L\otimes L' \to 0 \]

\[ \zeta : \underline{\Phi}_B \Longrightarrow \underline{L}_P \]

\[ \varphi_B \longmapsto \bigl( x\otimes y \mapsto \varphi_B(x,y) \bigr) \]

\[ P \xrightarrow{\ \Delta_{H,H'}\ } \det H \otimes \det H' \]

\[ P \simeq \underline{\mathrm{Hom}}(\check{H},H') \to \underline{\mathrm{Hom}}(\det\check{H},\det H') \]

\[ \underline{L}_P \subset \mathrm{Bil}(H,H';k) . \]

\[ \Delta_P,\ T_P,\ \varphi_P,\ \psi_P,\ \Delta_B,\ \psi_B,\ \varphi_B \]

\[ \operatorname{disc}'(\Delta_B) \in (\det B)^{\otimes-2} = (L\otimes L')^{\otimes-2} \]

\[ \underline{L}_P \xrightarrow{\ \delta_{P,B}\ } (L\otimes L')^{\otimes-2} \]

\[ \underbrace{(L\otimes L')^{-1}}_{\text{esp. des translations de } \underline{L}_P} \longrightarrow (L\otimes L')^{-2} \]

\[ \boxed{\, X\times_S X' \xrightarrow{\ \alpha\ } \tilde{S} \,} \]

\[ \alpha(\bar{x},x') = \alpha(x,\bar{x}') = \overline{\alpha(x,x')} \]

\[ X\wedge_S X' \longrightarrow S , \]

\[ \varphi : H\to k , \qquad \varphi' : H'\to k \]

\[ \text{\struck{$\varphi\otimes\varphi' : H\otimes H' = P\to k \qquad u\otimes v\mapsto \varphi(u)\varphi'(v)$}} \]

\[ \varphi(u) = z , \qquad \varphi'(v) = z' \]

\[ z^2 - bz + c = 0 \qquad z'^2 - b'z' + c' = 0 , \]

\[ t^2 - bb't + (b^2b' + b'^2b - 4cc') = 0 . \]

\[ \Delta = (bb')^2 - 4(b^2c' + b'^2c - 4cc') = \underbrace{(b^2-4c)}_{\delta}\,\underbrace{(b'^2-4c')}_{\delta'} \]

\[ 2z = b\pm\sqrt{\delta} , \qquad 2z' = b'\pm\sqrt{\delta'} \]

\[ \delta = (2z-b)^2 \qquad \delta' = (2z'-b)^2 \]

\[ \tfrac14 (2t-bb')^2 = \Delta = \delta\delta' = \bigl((2z-b)(2z'-b')\bigr)^2 \]

\[ 2t - bb' = \pm(2z-b)(2z'-b') \]

\[ 2t = bb' \pm (2z-b)(2z'-b') . \]

\[ 2t = -4zz' + 2(b'z+bz') \]

\[ \boxed{\, t = (b'z+bz') - 2zz' \,} \]

\[ t' = \bigl(bb' - (b'z+bz')\bigr) + 2zz' \qquad (= bb'-t) \]

\[ tt' = b^2c' + b'^2c - 4cc' \]

\[ \boxed{\, t_{\varphi,\varphi'}(u,v) = \operatorname{Tr}(v)\varphi(u) + \operatorname{Tr}(u)\varphi'(v) - 2\varphi(u)\varphi'(v) \,} \]

\[ t(u,1) = 2\varphi(u) + \operatorname{Tr}(u) - 2\varphi(u) = \operatorname{Tr}(u) \]

\[ t(u,1) = \operatorname{Tr} u , \qquad t(1,v) = \operatorname{Tr} v \]

\[ t_{\varphi,\varphi'} \in \underline{L}_P \]

\[ \delta(t_{\varphi,\varphi'}) = 0 \]

\[ \varphi(u) = \varphi(v) = 0 \]

\[ t(u,v) = 0 . \]

\[ \delta(u,v,t) = t^2 - bb't + (b^2c' + b'^2c - 4cc') = t(t-bb') \quad \text{car } c=c'=0 \]

\[ X\times X' \longrightarrow X\wedge X' \qquad (\overset{\text{déf}}{=} \tilde{S}) \]

\[ T \longmapsto T' \]

\[ T'(u,v) = T(u',v) = T(u,v') \]

\[ \boxed{\, T(u',v) = T(u,v') \,} \overset{\text{déf}}{=} T'(u,v) \]

\[ \operatorname{Tr}_{A_T}(u'v) = \operatorname{Tr}_{A_T}(uv') \]

\[ (T')' = T , \]

\[
  \delta(T) = \delta(T')
\]

\[
  \widetilde{S} = X \wedge X' = \delta^{-1}\bigl(\text{section nulle de } \mathbb{V}(L \otimes L')\bigr).
\]

\[
  t \longmapsto bb' - t ,
\]

\[
  X \times_S X' \longrightarrow X \wedge_S X'
\]

\[
  X \wedge_S X' \simeq X \times_S X' \big/ \underbrace{\sigma_X \times_S \sigma_{X'}}_{= \sigma_P}
\]

\[
  \boxed{\; u * v \overset{\mathrm{def}}{=} \operatorname{Tr}(v)\,u + \operatorname{Tr}(u)\,v - 2\underbrace{uv}_{= u \otimes v} \in (H \otimes H')^{\sigma_P} \;}
\]

\[
  \left\{
  \begin{aligned}
    \operatorname{Tr}(u * v) &= bb' \\
    \det(u * v) &= \delta(u, v) = b^2c + b'^2c' - 4cc'
  \end{aligned}
  \right.
\]

\[
  \boxed{\; T_{\varphi,\varphi'}(u,v) = \varphi \otimes \varphi'(u * v) \;}
\]

\[
  \underset{\text{discr}}{\delta(u * v)} = \delta(u)\,\delta(v)
\]

\[
  \left\{
  \begin{aligned}
    (u * v)' &= u' * v = u * v' \\
    u' * v' &= u * v
  \end{aligned}
  \right.
\]

\[
  \delta : L_P \longrightarrow (L \otimes L')^{-1}
\]

\[
  k + k\cdot v + kw
\]

\[
  H = k \oplus L, \qquad H' = k \oplus L', \qquad
  P = H \otimes H' = \underbrace{k \oplus L \oplus L'}_{} + L \otimes L'
\]

\[
  \delta_{H,H'} : \underset{\displaystyle\simeq\ (L \otimes L')^{-1}}{T_{H,H'}} \longrightarrow (L \otimes L')^{\otimes -2}
\]

\[
  (L \otimes L')^{-1} \xrightarrow[\text{élévation au carré}]{} (L \otimes L')^{\otimes -2}
\]

\[
  H \wedge H' \simeq k \oplus (L \otimes L') \qquad \text{($L \otimes L'$ formé d'élts de carré nul)}
\]

\[
  X \times_S X' \longrightarrow X'' = X \wedge X'
\]

\[
  \underset{\substack{\text{sections de } L, L' \\ \text{de carré nul}}}{(\xi, \eta)} \longrightarrow \underset{\substack{\text{section de } L \otimes L' \\ \text{de carré nul}}}{\xi \otimes \eta}
\]

\[
  H \otimes H' \simeq k \oplus L \oplus L' \oplus L \otimes L' \longleftarrow H'' = k \oplus L \otimes L'
\]

\[
  H, \quad H', \quad H''
\]

\[
  \sigma_H, \quad \sigma_{H'} \quad \text{et} \quad \sigma_{H''}
\]

\[
  H = P^{\sigma_H}, \quad H' = P^{\sigma_{H'}}, \quad H'' = P^{\sigma_{H''}}, \qquad
  H \cap H' = H' \cap H'' = H'' \cap H = k \quad (\text{\uncertain{union} !})
\]

\[
  H'' \simeq H \wedge H' \iff H' \simeq H \wedge H''
\]

\[
  u^2 = u, \quad v^2 = 0 \qquad (\text{donc } b = 1,\ c = b' = c' = 0)
\]

\[
  w = u * v = v - 2uv
\]

\[
  1, \quad u, \quad v, \quad w
\]

\[
  v^2 = w^2 = vw = 0
\]

\[
  u \underset{L,L'}{\otimes} v \longmapsto u * v = -2uv
\]

\[
  X_L \wedge X_{L'} \simeq X_{L \otimes L'} .
\]

\[
  X_{L''} \simeq X_L \wedge X_{L'} \qquad \text{i.e.\ } L'' \simeq L \otimes L'
\]

\[
  X_L \wedge X_L \simeq \struck{\ill{}} X_{L^{\otimes 2}} \simeq k + L^{\otimes 2}
\]

\[
  \bigwedge_{i \in I} X_i \qquad I \text{ un ens.\ fini}
\]

\[
  \begin{aligned}
    f(1, u_2, u_3) &= \operatorname{Tr}(u_2)\operatorname{Tr}(u_3) \\
    f(u_1, 1, u_3) &= \operatorname{Tr}(u_1)\operatorname{Tr}(u_3) \\
    f(1, u_2, u_3) &= \operatorname{Tr}(u_2)\operatorname{Tr}(u_3)
  \end{aligned}
\]

\[
  x + x' \in k.1 ,
\]

\[
  x' = T(x).1 - x \qquad \text{alors } x \mapsto x' \text{ est involution}
\]

\[
  E \xrightarrow{\ \varepsilon\ } k
\]

\[
  t \in E \quad \text{avec} \quad \varepsilon(t) = 2 ,
\]

\[
  x \longmapsto \varepsilon(x)\,t - x
\]

\[
  f : A_1 \times A_2 \longrightarrow B
\]

\[
  \text{\struck{$f(x$}} \quad \exists\, f^0 : A_1^0 \times A_2^0 \longrightarrow B^0 \quad \text{bil}
\]

\[
  (*) \quad
  \left\{
  \begin{aligned}
    f(x_1 + \alpha_1, x_2) &= f(x_1, x_2) + f^0(\alpha_1, \sigma_2 x_2 - x_2) \\
    f(x_1, x_2 + \alpha_2) &= f(x_1, x_2) + f^0(\sigma_1 x_1 - x_1, \alpha_2)
  \end{aligned}
  \right.
\]

\[
  \begin{aligned}
    f(x_1 + \alpha_1, x_2 + \alpha_2) = f(x_1, x_2) &+ f^0(\alpha_1, \sigma_2 x_2 - x_2) + f^0(\sigma_1 x_1 - x_1, \alpha_1) \\
    &- 2 f^0(\alpha_1, \alpha_2)
  \end{aligned}
\]

\[
  \begin{aligned}
    \sigma_1 x_1 - x_1 &= t_1 - 2x_1 = \tau_1 \in A_1^0 \\
    \sigma_2 x_2 - x_2 &= t_2 - 2x_2 = \tau_2 \in A_2^0
  \end{aligned}
\]

\[
  \underset{\overset{\| \mathrm{df}}{f(x_1 + \alpha_1,\, x_2 + \alpha_2)}}{g(\alpha_1, \alpha_2)}
  = \underset{\overset{\|}{f(x_1, x_2)}}{c} + f^0(\alpha_1, \tau_2) + f^0(\tau_1, \alpha_2) - 2 f^0(\alpha_1, \alpha_2)
\]

\[
  (f, f^0) : \quad
  \begin{aligned}
    f &: A_1 \times A_2 \longrightarrow B \\
    f^0 &: A_1^0 \times A_2^0 \longrightarrow B^0
  \end{aligned}
\]

\[
  A_1^0 \otimes A_2^0
\]

\[
  (x_1, x_2) \longmapsto x_1 * x_2 : A_1 \times A_2 \longrightarrow A_1 * A_2
\]

\[
  E_1 \otimes E_2 \supset A_1^0 \otimes A_2^0
\]

\[
  \boxed{\;
  \begin{aligned}
    (x_1, x_2) \longmapsto x_1 * x_2 &= t_1 \otimes x_2 + x_1 \otimes t_2 - 2\, x_1 \otimes x_2 \\
    &= \sigma_1(x_1) \otimes x_2 + x_1 \otimes \sigma(x_2)
  \end{aligned}
  \;}
\]

\[
  \begin{aligned}
    f^0_{A_1, A_2} : A_1^0 \times A_2^0 &\longrightarrow A_1^0 \otimes A_2^0 \\
    (\alpha_1, \alpha_2) &\longmapsto \alpha_1 \otimes \alpha_2
  \end{aligned}
\]

\[
  x_1 * x_2 \longmapsto \sigma_1(x_1) * x_2 \qquad x_1 * x_2 \longmapsto x_1 * \sigma_2(x_2)
\]

\[
  \underset{t_1 - x_1}{\underbrace{\sigma_1(x_1)}} * x_2 = x_1 * \underset{t_2 - x_2}{\underbrace{\sigma_2(x_2)}} = t_1 \otimes t_2 - x_1 * x_2
\]

\[
  f(\sigma_1(x_1), x_2) = f(x_1, \sigma_2(x_2)) = \sigma_B f(x_1, x_2) .
\]

\[
  k \times k \xrightarrow{\ \varepsilon\ } k , \qquad t_0 = (1,1) \mapsto 2
\]

\[
  \begin{aligned}
    k &\hookrightarrow k \times k \\
    \lambda &\longmapsto (\lambda, 1 - \lambda)
  \end{aligned}
  \qquad
  \begin{aligned}
    k &\longleftarrow k \times k \\
    \lambda &\longmapsto (\lambda, -\lambda)
  \end{aligned}
\]

\[
  \sigma_0(e_0) - e_0 = 1 \in k = A_0^0
\]

\[
  \begin{aligned}
    A &\xrightarrow{\ \sim\ } A_0 * A \\
    x &\longmapsto e_0 * x
  \end{aligned}
\]

\[
  A^0 \xrightarrow{\ \sim\ } \underset{\overset{\|}{A_0^0}}{k} \otimes A^0 \qquad \text{iso.\ can.}
\]

\[
  A_1 * A_2 * \cdots * A_n
\]

\[
  \begin{aligned}
    f &: A_1 \times \cdots \times A_n \longrightarrow B \\
    f^0 &: A_1^0 \times \cdots \times A_n^0 \longrightarrow B^0
  \end{aligned}
\]

\[
  \begin{aligned}
    f(x_1, \ldots, x_i + \alpha_i, \ldots, x_n) = f(x_1, \ldots, x_i, \ldots, x_n) \\
    + f^0(\sigma_1 x_1 - x_1, \ldots, \alpha_i, \ldots, \sigma_n x_n - x_n)
  \end{aligned}
\]

\[
  E_1 \otimes \cdots \otimes E_n
\]

\[
  x_1 * x_2 * \cdots * x_n \qquad (\text{pour } x_i \in A_i)
\]

\[
  \begin{aligned}
    x_1 * x_2 * \cdots * x_n = \Bigl(\sum t_1 \cdots t_{n-1} x_n\Bigr) - 2\Bigl(\sum t_1 \cdots t_{n-2} x_{n-1} x_n\Bigr) \\
    + 4\Bigl(\sum t_1 \cdots t_{n-3} x_{n-2} x_{n-1} x_n\Bigr) + \cdots + (-1)^{n-1} 2^{n-1} x_1 x_2 \cdots x_n .
  \end{aligned}
\]

\[
  t_1 t_2 \cdots t_n - x_1 * x_2 * \cdots * x_n .
\]

\[
  \begin{aligned}
    E_1 \times \cdots \times E_n &\longrightarrow E_1 \otimes \cdots \otimes E_n \\
    (x_1, \ldots, x_n) &\longmapsto x_1 * x_2 * \cdots * x_n
  \end{aligned}
\]

\[
  \mathop{\ast}_i x_i = \sum_{i=1}^{n} (-1)^{i-1} 2^{i-1}
  \Bigl(\sum \underbrace{\varepsilon_1(x_1) t_1\, \varepsilon_2(x_2) t_2 \cdots \varepsilon_{n-i}(x_{n-i}) t_{n-i}}_{n-i \text{ facteurs } \varepsilon(x_j) t_j}\;
  \underbrace{x_{n-i+1} \cdots x_n}_{i \text{ facteurs}}\Bigr).
\]

\[
  \begin{aligned}
    x_1 * x_2 * \cdots * x_n &= \sum_{\substack{i \text{ impair} \\ 1 \leq i \leq n}} \ \sum_{\binom{n}{i} \text{ termes}} x_1 \cdots x_i\, \sigma_{i+1}(x_{i+1}) \cdots \sigma_n(x_n) && (2^{n-1} \text{ termes}) \\
    \sigma(x_1 * \cdots * x_n) &= \sum_{\substack{i \text{ pair} \\ 0 \leq i \leq n}} \ \sum_{\binom{n}{i} \text{ termes}} x_1 \cdots x_i\, \sigma_{i+1}(x_{i+1}) \cdots \sigma_n(x_n) && (2^{n-1} \text{ termes})
  \end{aligned}
\]

\[
  (x_1 * \cdots * x_n) + \sigma(x_1 * \cdots * x_n) = t_1 t_2 \cdots t_n\, \varepsilon_1(x_1) \varepsilon_2(x_2) \cdots \varepsilon_n(x_n)
\]

\[
  (x_1 + \alpha_1) * x_2 * \cdots * x_n = \alpha_1 \otimes (\sigma_2(x_2) - x_2) \otimes \cdots \otimes (\sigma_n(x_n) - x_n)
\]

\[
  \varepsilon : E \longrightarrow k
\]

\[
  \varepsilon(\underbrace{t_1 t_2 \cdots t_n}_{t}) = 2 .
\]

\[
  \varepsilon_1 \otimes \cdots \otimes \varepsilon_n : \bigotimes E_i \longrightarrow k
\]

\[
  P_\alpha = \bigl\{ x \in P \bigm| \sigma_1^{\beta_1} \otimes \cdots \otimes \sigma_n^{\beta_n}(x) = (-1)^{\alpha_1\beta_1 + \alpha_2\beta_2 + \cdots + \alpha_n\beta_n} x \bigr\}
  \quad \forall \text{ syst.\ d'entiers } \beta_i \bmod 2 .
\]

\[
  E = P_{1, \ldots, 1} = \bigl\{ x \in P \bigm| (\sigma_1^{\beta_1} \otimes \cdots \otimes \sigma_n^{\beta_n})(x) = (-1)^{\sum \beta_i} x \bigr\}
\]

\[
  \begin{aligned}
    E_1 \times \cdots \times E_n &\xrightarrow{\ f\ } E \\
    E_1^0 \times \cdots \times E_n^0 &\xrightarrow{\ f^0\ } E
  \end{aligned}
\]

\[
  \begin{aligned}
    f(x_1 + \alpha_1, x_2, \ldots, x_n) = f(x_1, \ldots, x_n) + f^0(\alpha_1, \sigma_2(x_2) - x_2, \ldots \\
    \ldots, \sigma_n(x_n) - x_n)
  \end{aligned}
\]

\[
  f(\alpha_1, \alpha_2, \ldots, \alpha_n) = (-2)^{n-1} f^0(\alpha_1, \alpha_2, \ldots, \alpha_n)
\]

\[
  0 \longrightarrow \bigotimes E_i^0 \xrightarrow{\ i\ } \mathop{\ast}_i (E_i, \sigma_i)^{\wedge} \xrightarrow{\ J\ } \bigotimes (E_i/E_i^0) \longrightarrow 0
\]

\[
  \mathop{\ast}_i (E_i, \sigma_i) \xrightarrow{\ \tau\ } \bigotimes E_i
\]

\[
  q_E : E \longrightarrow L^{\otimes 2}
\]

\[
  L \longrightarrow L^{\otimes 2}, \qquad u \longmapsto u^{\otimes 2}
\]

\[
  \varphi_{q_E}(u, x) = u \otimes (x \wedge t) \in L^{\otimes 2},
  \qquad u \in L,\; x \in E,\; x\wedge t \in \det E \simeq L .
\]

\[
  \xi \longmapsto q_\xi, \qquad A \ast A \longrightarrow
  \left\{\text{formes quadratiques sur } E \text{ qui sont compatibles à } t\right\}
\]

\[
  A^{\otimes 2} \longrightarrow L^{\otimes 2}, \qquad \alpha \longmapsto -\alpha
\]

\[
  \xi + q_\xi(x) = x \ast x .
\]

\[
  \underbrace{\mathrm{Quadcomp}(E, \varepsilon, t)}_{\text{sur } (L,\sigma)} \simeq L \mathbin{\ast_\sigma} L
\]

\[
  \sigma_L(x) = \tau - x
\]

\[
  A \longrightarrow A \ast A
\]

\[
  x \longmapsto x \ast x .
\]

\[
  \xi \in A \ast A,
\]

\[
  q_\xi : A \longrightarrow L^{\otimes 2}
\]

\[
  \xi + q_\xi(x) = x \ast x \qquad \forall x \text{ dans } A
\]

\[
  (\xi + \alpha) + q_{\xi+\alpha}(x) = x \ast x
\]

\[
  q_{\xi+\alpha}(x) = q_\xi(x) - \alpha
  \qquad \text{pour } x \in A, \text{ i.e. } \varepsilon(x) = 1
\]

\[
  A \longrightarrow Q
\]

\[
  u \longmapsto u^{\otimes 2} : L \longrightarrow L^{\otimes 2},
\]

\[
  A \longrightarrow Q \longrightarrow L^{\otimes 2}
\]

\[
  \Lambda \times A \longrightarrow L^{\otimes 2}, \qquad (q, x) \longmapsto q(x)
\]

\[
  (q + \alpha, x) \longmapsto q(x) + \alpha \qquad \alpha \in L^{\otimes 2}
\]

\[
  Q_\sigma = \mathrm{Hom}_{\text{torseurs inversés}}(\Lambda, L^{\otimes 2})
\]

\[
  \Lambda \simeq \mathrm{Hom}_{\text{torseurs sous } L^{\otimes 2}}(Q, L^{\otimes 2})
\]

\[
  A \longrightarrow Q_\sigma
\]

\[
  q : A \longrightarrow Q
\]

\[
  q(x + \alpha) = q(x) \mathbin{\dot{-}} (t \wedge x) \otimes \alpha + \alpha \otimes \alpha ,
  \qquad \alpha \in L,\ t \wedge x \in L .
\]

\[
  Q(A, \sigma) = Q(A)
\]

\[
  A = \text{\struck{$Q(A_1$}}\ A_1 \ast A_2 \qquad \text{tors.\ sous } L_1 \otimes L_2 .
\]

\[
  \underbrace{Q(A)}_{\text{sous } L^{\otimes 2}} \simeq
  \underbrace{Q(A_1)}_{\text{sous } L_1^{\otimes 2}} \ast
  \underbrace{Q(A_2)}_{\text{sous } L_2^{\otimes 2}}
  \qquad \text{(iso.\ canonique)}
\]

\[
  Q(A_1) \times Q(A_2) \longrightarrow Q(A),
  \qquad q_1, q_2 \longmapsto q_1 \otimes q_2 \,|\, A_1 \ast A_2 \subset E_1 \otimes E_2
\]

\[
  A_i \simeq A_i^{\circ}, \qquad \sigma_i \text{ devient } x \longmapsto b_i - x
\]

\[
  q_{c_i} \text{ ou } q_{b_i, c_i} : x_i \longmapsto x_i^2 - b_i x_i + c_i ,
  \qquad c_i \in L^{\otimes 2} \text{ \uncertain{comme}}
\]

\[
  q_{c_1} \otimes q_{c_2} = q_c :
  x \longmapsto x^2 - (b_1 b_2) x + (b_2^2 c_1 + b_1^2 c_2 - 4 c_1 c_2) ,
  \qquad x \in L_1 \otimes L_2 .
\]

\[
  q_{c_1} \otimes q_{c_2} = q_c \qquad c = b_2^2 c_1 + b_1^2 c_2 - 4 c_1 c_2
\]

\[
  (q_1 + \alpha) \otimes q_2 = q_1 \otimes q_2 + \ddot{\alpha}\, \delta_2(q_2),
  \qquad \alpha \in L_1^{\otimes 2},\ \delta_2(q_2) \in L_2^{\otimes 2}
\]

\[
  q_1 \otimes (q_2 + \beta) = q_1 \otimes q_2 + \delta_1(q_1)\, \beta
\]

\[
  \delta_1(q_{c_1}) = b_1^2 - 4 c_1, \qquad \delta_2(q_{c_2}) = b_2^2 - 4 c_2 .
\]

\[
  \delta(q_1 \otimes q_2) = \delta(q_1)\, \delta(q_2),
  \qquad \delta(q_1) \in L_1^{\otimes 2},\ \delta(q_2) \in L_2^{\otimes 2},
\]

\begin{align*}
  x_1 \ast x_2 &= (o_1 + x_1) \ast (o_2 + x_2)
    = o_1 \ast o_2 + \bigl(x_1 \otimes (\overbrace{\sigma_2(o_2) - o_2}^{b_2})\bigr) \\
  &\qquad + (\sigma_1(o_1) - o_1) \otimes x_2 - 2\, x_1 \otimes x_2 \\
  &= o + \underbrace{x_1 \otimes b_2 + b_1 \otimes x_2 - 2\, x_1 x_2}_{x_1 b_2 + b_1 x_2 - 2 x_1 x_2}
\end{align*}

\[
  \pi_1(x_1) = q_{x_1(b - x_1)} =
  \bigl(z_1 \mapsto z_1^2 - b_1 z_1 + x_1(b_1 - x_1) = (z_1 - x_1)(z_1 - (b_1 - x_1))\bigr)
\]

\begin{align*}
  z \longmapsto{}& z^2 - b_1 b_2 z + b_2^2 x_1(b_1 - x_1) + b_1^2 x_2 (b_2 - x_2) \\
  &- 4 x_1 (b_1 - x_1) x_2 (b_2 - x_2)
\end{align*}

\begin{align*}
  \pi(x_1 \ast x_2) &= \bigl(z \mapsto z^2 - \underbrace{b_1 b_2}_{b} z
     + \underbrace{(x_1 \ast x_2)(b_1 b_2 - x_1 \ast x_2)}\bigr) \\
  &\qquad (x_1 b_2 + b_1 x_2 - 2 x_1 x_2)(b_1 b_2 - x_1 b_2 - b_1 x_2 + 2 x_1 x_2)
\end{align*}

\[
  Q(A_1 \ast A_2) \simeq Q(A_1) \ast Q(A_2)
\]

\[
  0 \longrightarrow E^{\circ} \longrightarrow E \xrightarrow{\ \varepsilon\ } F \longrightarrow 0,
  \qquad F \simeq E/E^{\circ}
\]

\[
  u(x) = x + \sigma(x) = 0 \quad \text{si } x \in E^{\circ}
\]

\[
  \pi : F \longrightarrow E \qquad u = \pi \circ \varepsilon
\]

\[
  \underbrace{\sigma(x)}_{u(x)} = \pi(\varepsilon(x)) - x .
\]

\[
  x \equiv \pi \varepsilon(x) - x \quad (E^{\circ}) \qquad \forall x \in E
\]

\[
  \pi \underbrace{\varepsilon(x)}_{y} \equiv 2x \qquad \forall x \in E
\]

\[
  \boxed{\varepsilon \pi(y) = 2y}
\]

\[
  \pi : F \longrightarrow E \quad \text{telle que} \quad \varepsilon \pi = \lambda\, \mathrm{id}_F ,
\]

\[
  \varepsilon(t) = \lambda .
\]

\[
  \varepsilon^{-1}(\lambda) = {}^{\lambda}A
\]

\[
  x \longmapsto \lambda x \qquad A \longrightarrow {}^{\lambda}A
\]

\[
  {}^{\lambda}A \simeq A^{\circ}, \qquad x \longmapsto x - t
\]

\[
  u : A \longrightarrow A^{\circ} \qquad u(x) = \lambda x - t
\]

\[
  A^{\circ} \xrightarrow{\ \lambda\,\mathrm{id}_{A^{\circ}}\ } A^{\circ}
\]

\[
  u(x) = 2x - t = -(\sigma(x) - x),
\]

\[
  \sigma(x) - x .
\]

\[
  E \xrightarrow{\ u\ } A^{\circ} \qquad u(x) = \lambda x - \underbrace{\varepsilon(x)\, t}_{= \pi \varepsilon(x)}
\]

\[
  0 \longrightarrow E^{\circ} \longrightarrow E \longrightarrow F \longrightarrow 0,
  \qquad F \simeq E/E^{\circ}
\]

\[
  \pi : F \longrightarrow E ,
\]

\begin{gather*}
  u = u^{\pi} : E \longrightarrow E^{\circ} \\
  u^{\pi}(x) = \lambda x - \pi \varepsilon(x)
\end{gather*}

\[
  \varepsilon u(x) = \lambda \underbrace{\varepsilon(x)}_{y} - \varepsilon\pi(\varepsilon(x))
  = \lambda y - \underbrace{\varepsilon\pi(y)}_{\lambda y} = 0
\]

\[
  u | E^{\circ} = \lambda\, \mathrm{id}_{E^{\circ}} \qquad \text{i.e. } u(x) = \lambda x \text{ si } x \in E^{\circ}
\]

\[
  \pi(\varepsilon(x)) = \lambda x - u(x)
\]

\[
  0 \longrightarrow E_i^{\circ} \xrightarrow{\ i_{E_i}\ } E_i \longrightarrow F_i \longrightarrow 0
  \qquad u_i \circ i_{E_i} = \lambda\, \mathrm{id}_{E_i^{\circ}}
\]

\[
  \textstyle\prod E_i \xrightarrow{\ f\ } B \qquad \prod E_i^{\circ} \xrightarrow{\ f^{\circ}\ } B
\]

\begin{align*}
  f(x_1, \ldots, x_i + \alpha_i, \ldots, x_n) &= f(x_1, \ldots, x_i, \ldots, x_n) \\
  &\quad + f^{\circ}(u_1(x_1), \ldots, \alpha_i, \ldots, u_n(x_n))
\end{align*}

\[
  f(\alpha_1, \ldots, \alpha_i, \ldots, \alpha_n) = \lambda^{n-1} f^{\circ}(\alpha_1, \ldots, \alpha_i, \ldots, \alpha_n)
\]

\[
  \mathop{\ast}_{i,\lambda} E_i \qquad (E_1 \mathbin{\ast_\lambda} E_2 \text{ etc.})
\]

\[
  \textstyle\bigotimes_i E_i \oplus \bigotimes_i E_i^{\circ}
\]

\[
  \textstyle\bigotimes_i E_i^{\circ} \longrightarrow \mathop{\ast}_{i,\lambda} E_i \longrightarrow \bigotimes_i F_i \longrightarrow 0
\]

\[
  E_i \simeq E_i^{\circ} \oplus F_i
\]

\[
  t_i : F_i \longrightarrow E_i^{\circ}
\]

\[
  \mathop{\ast}_{i} E_i \longrightarrow \textstyle\bigotimes E_i^{\circ}
\]

\[
  f = \textstyle\bigotimes u_i : \bigotimes E_i \longrightarrow \bigotimes E_i^{\circ}
\]

\[
  f(x_1, \ldots, x_n) = u_1(x_1) \otimes \cdots \otimes u_n(x_n)
\]

\begin{align*}
  f(x_1 + \alpha_1, x_2, \ldots, x_n) &= (u_1(x_1) + \lambda\alpha_1) \otimes u_2(x_2) \otimes \cdots \otimes u_n(x_n) \\
  &= f(x_1, \ldots, x_n) + \underbrace{\lambda\alpha_1}_{\text{\struck{\ill{}}}} \otimes u_2(x_2) \otimes \cdots \otimes u_n(x_n)
\end{align*}

\[
  f^{\circ}(\alpha_1, \ldots, \alpha_n) = \lambda\, \alpha_1 \otimes \cdots \otimes \alpha_n
\]

\begin{align*}
  f(\underbrace{x_1 + \alpha_1}_{X_1}, \underbrace{x_2 + \alpha_2}_{X_2})
  &= g(x_1, x_2) + \text{\struck{\ill{}}}\, f^{\circ}(\alpha_1, b_2(x_2)) \\
  &\quad + f^{\circ}(b_1(x_1), \alpha_2) + \lambda \underbrace{f^{\circ}(\alpha_1, \alpha_2)}_{\substack{\shortparallel \\ f^{\circ}(u_1(\alpha_1), \alpha_2) \\ = f^{\circ}(\alpha_1, u_2(\alpha_2))}}
\end{align*}

\begin{align*}
  f(X_1 + \alpha'_1, X_2) &= f(X_1, X_2) + f^{\circ}(\alpha'_1, \underbrace{b_2(x_2)}) \\
  &\quad + \lambda f^{\circ}(\alpha'_1, \alpha_2)
\end{align*}

\[
  = f(X_1, X_2) + f^{\circ}(\alpha'_1, \underbrace{u_2(X_2)}_{b_2(x_2) + \lambda\alpha_2})
  \qquad \text{OK.}
\]

\[
  f(x_1, \ldots, x_n) = f^{\circ}(\alpha_1, u_2(x_2), \ldots, u_n(x_n)) .
\]

\[
  0 \longrightarrow E^{\circ} \longrightarrow E \longrightarrow F \longrightarrow 0,
\]

\[
  0 \longrightarrow k \longrightarrow E \longrightarrow k \longrightarrow 0
\]

\[
  E_\beta \mathbin{\ast_\lambda} E_2 \simeq \beta E_2 ,
\]

\[
  \det(x+y) = \det x + \det y + \mathrm{Tr}\, x\, \mathrm{Tr}\, y - \mathrm{Tr}\, xy ,
\]

\[
  \varphi_{\det}(x,y) = \mathrm{Tr}\, x\, \mathrm{Tr}\, y - \underbrace{\mathrm{Tr}\, xy}_{\varphi(x,y)} ,
\]

\[
  \mathrm{Tr}\, xyz \qquad \forall z \in \mathcal{A} .
\]

\[
  x \longmapsto x' = \mathrm{Tr}(x).1 - x .
\]

\[
  \begin{aligned}
    \tau &= \mathrm{Tr}\, xyz \quad (= \mathrm{Tr}\, yzx = \mathrm{Tr}\, zxy) \\
    \tau' &= \mathrm{Tr}(zyx) \quad (= \mathrm{Tr}(yxz) = \mathrm{Tr}(xzy))
  \end{aligned}
\]

\[
  \tau + \tau' = \mathrm{Tr}\,(xy+yx)z
\]

\[
  xy + yx = \underbrace{\mathrm{Tr}(y)}_{b'} x + \underbrace{\mathrm{Tr}(x)}_{b} y
  + \bigl(\underbrace{\mathrm{Tr}(xy)}_{t''} - \underbrace{\mathrm{Tr}\, x}_{b}\, \underbrace{\mathrm{Tr}\, y}_{b'}\bigr) 1
\]

\[
  \text{\struck{$(xy)z+$}}\quad (xy+yx)z = b'xz + byz + (t''-bb')z
\]

\[
  \boxed{\tau + \tau' = bt + b't' + b''t'' - bb'b''}
  \overset{\text{déf}}{=} \Sigma(b,b',b'',t,t',t'')
\]

\[
  \mathrm{Tr}(xyz) = - \mathrm{Tr}(yxz) .
\]

\[
  z' = z - \alpha 1 - \beta x - \gamma y
\]

\[
  \begin{aligned}
    &b'' - 2\alpha - b\beta - b'\gamma = 0 \\
    &t' - b\alpha + (2c - b^2)\beta - t''\gamma = 0 \\
    &t - b'\alpha - t''\beta + (2c' - b'^2)\gamma = 0
  \end{aligned}
\]

\[
  \left\{
  \begin{aligned}
    2\alpha + b\beta + b'\gamma &= b'' \\
    b\alpha + (b^2 - 2c)\beta + t''\gamma &= t' \\
    b'\alpha + t''\beta + (b'^2 - 2c')\gamma &= t
  \end{aligned}
  \right.
\]

\[
  \begin{pmatrix}
    2 & b & b' \\
    b & b^2 - 2c & t'' \\
    b' & t'' & b'^2 - 2c'
  \end{pmatrix}
\]

\[
  (1 \wedge \text{\struck{$x$}}\, x \wedge y \wedge z) \otimes e^{-1} = 2\varepsilon_0
  \qquad \text{où} \quad
  \varepsilon_0 = (\alpha \wedge \alpha' \wedge x_0 \wedge y_0) \otimes e^{-1} \in \pm 1 ,
\]

\[
  \tau - \tau' = \mathrm{Tr}\, \underbrace{[x,y]}_{z} . z = \mathrm{Tr}\, z^2 = \mathrm{Tr}\, 1 = 2
\]

\[
  \Delta = 2\varepsilon\varepsilon_0 \quad \text{i.e.} \quad \varepsilon = \tfrac12\, \varepsilon_0
\]

\[
  \boxed{\tau - \tau' = \tfrac12\, \varepsilon_0\, \underbrace{(1 \wedge x \wedge y \wedge z) \otimes e^{-1}}_{\overset{\text{déf}}{=}\ \Delta}}
  \qquad \text{i.e.} \quad 2(\tau - \tau') = \varepsilon_0 \Delta
\]

\[
  4\tau\tau' = (\tau+\tau')^2 - (\tau-\tau')^2 = \Sigma^2 - \tfrac14 \Delta^2
\]

\[
  \Sigma^2 - \tfrac14 \Delta^2 \in 4\, \mathbb{Z}[b,b',b'',t,t',t'']
\]

\[
  \text{\struck{$4\Sigma^2 - \Delta^2 \in 16\, \mathbb{Z}[b,b',b'',t,t',t'']$}}
\]

\[
  \tau\tau' = \Pi(b,b',b'',t,t',t'') \in \mathbb{Z}[b,b',b'',t,t',t'']
\]

\[
  \tau' - \tau = \mathrm{Tr}\, xyz - \mathrm{Tr}\, yxz = \mathrm{Tr}([x,y]z)
  = \mathrm{Tr}([y,z]x) = \mathrm{Tr}([z,x].y)
\]

\[
  \tau' - \tau = \text{\struck{$\ill{}$}}\, (a \wedge x \wedge y \wedge z) \otimes e^{-1}
\]

\[
  g(a) \wedge g(x) \wedge g(y) \wedge g(z) = a \wedge x \wedge y \wedge z
  \quad (= a \wedge g(x) \wedge g(y) \wedge g(z)) ,
\]

\[
  \tau' - \tau = \varepsilon\, (1 \wedge x \wedge y \wedge z) \otimes e^{-1}
\]

\[
  x_0 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad
  y_0 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad
  z_0 = [x_0, y_0] = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = 1 - 2\alpha
  \quad \text{où } \alpha = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
\]

\[
  1 \wedge x_0 \wedge y_0 \wedge z_0 = -2\, 1 \wedge x_0 \wedge y_0 \wedge \alpha
  = -2\, \alpha' \wedge x_0 \wedge y_0 \wedge \alpha
  = 2\, \alpha \wedge \alpha' \wedge x_0 \wedge y_0
\]

\[
  \alpha = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad
  \alpha' = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\]

\[
  \begin{aligned}
    \Delta^2 = \alpha\, tt't''
    &+ \beta\,(b^2t^2 + b'^2t'^2 + b''^2t''^2)
    + \gamma\,(b'b''t't'' + b''bt''t + bb'tt') \\
    &+ \delta\,(b'b''b^2 t + b''bb'^2 t' + bb'b''^2 t'')
    + \eta\, b^2 b'^2 b''^2
  \end{aligned}
\]

\[
  \alpha, \beta, \gamma, \delta \in \mathbb{Z}
\]

\[
  b, b', b'' = 0, \quad \text{et} \quad \Delta = 2\varepsilon_0, \quad \Delta^2 = 4 ,
\]

\[
  4 = \alpha\, tt't''
\]

\[
  t'' = \mathrm{Tr}\, x_0 y_0 = 1, \quad t = \mathrm{Tr}\, y_0 z_0 = 0, \quad t' = \mathrm{Tr}\, z_0 x_0 = 0
\]

\[
  \Delta(x,y,z)^2 = \det
  \begin{pmatrix}
    \varphi(1,1) & \varphi(1,x) & \varphi(1,y) & \varphi(1,z) \\
    \varphi(x,1) & \varphi(x,x) & \varphi(x,y) & \varphi(x,z) \\
    \varphi(y,1) & \varphi(y,x) & \varphi(y,y) & \varphi(y,z) \\
    \varphi(z,1) & \varphi(z,x) & \varphi(z,y) & \varphi(z,z)
  \end{pmatrix}
\]

\[
  \varphi(u,v) = \mathrm{Tr}\, u\, \mathrm{Tr}\, v - \mathrm{Tr}(uv) =
\]

\[
  \begin{aligned}
    \varphi(u,1) &= \mathrm{Tr}(u) \quad \text{\struck{$\ill{}$}} \\
    \varphi(u,u) &= 2 \det u
  \end{aligned}
\]

\[
  \boxed{\Delta(x,y,z)^2 = \det
  \begin{pmatrix}
    2 & b & b' & b'' \\
    b & 2c & bb' - t'' & bb'' - t' \\
    b' & bb' - t'' & 2c' & b'b'' - t \\
    b'' & bb'' - t' & b'b'' - t & 2c''
  \end{pmatrix}}
\]

\[
  4\tau\tau' = \Sigma^2 - \Delta^2
\]

\[
  \boxed{\Sigma = bt + b't' + b''t'' - bb'b''}
\]

\[
  \Sigma^2 - \Delta^2 = 4\Pi(b,b',b'',t,t',t'',c,c',c'')
\]

\[
  \boxed{
  \begin{aligned}
    \Pi = tt't'' &+ (ct^2 + c't'^2 + c''t''^2) + (b^2c'c'' + b'^2c''c + b''^2cc') \\
    &- (b'b''ct + b''bc't' + bb'c''t'')
  \end{aligned}}
\]

\[
  = tt't'' + ct(t - b'b'') + c't'(t' - b''b) + c''t''(t'' - bb')
  + (b^2c'c'' + b'^2c''c + b''^2cc')
\]

\[
  \left\{
  \begin{aligned}
    \tau + \tau' &= \Sigma \quad (= bt + b't' + b''t'' - bb'b'') \\
    \tau\tau' &= \Pi \quad \bigl(= tt't'' + (ct^2 + c't'^2 + c''t''^2) \\
    &\qquad\quad + (b^2c'c'' + b'^2c''c + b''^2cc') \\
    &\qquad\quad - (b'b''ct + b''bc't' + bb'c''t'')\bigr)
  \end{aligned}
  \right.
\]

\[
  \tau - \tau' = \varepsilon_0\, \Delta
\]

\[
  A = k[\underline{b}, \underline{b}', \underline{b}'', \underline{c}, \underline{c}', \underline{c}'',
  \underline{t}, \underline{t}', \underline{t}'']
\]

\[
  \left\{
  \begin{aligned}
    &\underline{b} = \mathrm{Tr}\, x, \quad \underline{b}' = \mathrm{Tr}\, y, \quad \underline{b}'' = \mathrm{Tr}\, z \\
    &\underline{c} = \det x, \quad \underline{c}' = \det y, \quad \underline{c}'' = \det \uncertain{y} \\
    &\underline{t} = \mathrm{Tr}\, yz, \quad \underline{t}' = \mathrm{Tr}\, zx, \quad \underline{t}'' = \mathrm{Tr}\, xy
  \end{aligned}
  \right.
\]

\[
  \tau = \mathrm{Tr}\, xyz \quad (= \mathrm{Tr}\, yzx = \mathrm{Tr}\, zxy)
\]

\[
  \boxed{\tau^2 - \Sigma\tau + \Pi = 0}
\]

\[
  B \simeq A[\underline{\tau}] / (\underline{\tau}^2 - \Sigma\underline{\tau} + \Pi) ,
\]

\[
  \tau \longmapsto \tau' = \Sigma - \tau ,
\]

\[
  \tau' = \mathrm{Tr}\, yxz = \mathrm{Tr}\, xzy = \mathrm{Tr}\, zyx .
\]

\[
  B = k[\underline{b}, \underline{b}', \underline{b}'', \underline{t}, \underline{t}', \underline{t}'',
  \underline{c}, \underline{c}', \underline{c}'', \underline{\tau}, \underline{\tau}']
  / (\underbrace{\underline{\tau} + \underline{\tau}' - \Sigma}, \underbrace{\tau\tau' - \Pi})
\]

\[
  \Delta = \tau - \tau'
\]

\[
  \Delta^2 = \Sigma^2 - 4\Pi \in A .
\]

\[
  \begin{aligned}
    \tau &= \tfrac12 (\Delta + \Sigma) \\
    \tau' &= \tfrac12 (-\Delta + \Sigma)
  \end{aligned}
\]

\[
  \Delta = \tau + \tau' = \Sigma \in A \quad (\text{en car.\ } 2)
\]

\[
  \left.
  \begin{aligned}
    b_i &= \mathrm{Tr}\, x_i \\
    c_i &= \det x_i
  \end{aligned}
  \right\} \quad 1 \leq i \leq 4
\]

\[
  t_{ij} = \text{\struck{$t$}}\, t_{\{i,j\}} = \mathrm{Tr}\, x_i x_j
  \qquad \{i,j\} \in \mathfrak{P}_2([1,4])
\]

\[
  t_{ijk} = \mathrm{Tr}(x_i x_j x_k)
  \qquad i, j, k \in [1,4] \text{ deux à deux distincts}
\]

\[
  t_{ijk} = t_{jki} = t_{kij}
\]

\[
  (*) \quad
  \left\{
  \begin{aligned}
    t_{ijk} + t_{jik} &= \Sigma_{\{i,j,k\}} \in \mathbb{Z}[b_i, c_i, t_i, b_j, c_j, t_j, b_k, c_k, t_k] \\
    t_{ijk} . t_{jik} &= \Pi_{\{i,j,k\}} \in \quad \text{id.}
  \end{aligned}
  \right.
\]

\[
  t_{2,3,4}, \quad t_{1,3,4}, \quad t_{1,2,4}, \quad t_{1,2,3} .
\]

\[
  \mathrm{Tr}\, x_1 x_2 x_3 x_4
\]

\[
  \beta' = -\beta
\]

\[
  \begin{aligned}
    &bc' + b'c'' + b''c \\
    &- (bc'' + b'c + b''c')
  \end{aligned}
\]

\[
  \alpha' = \qquad \overline{\alpha}' = \overline{\alpha} + \beta\sigma
\]

\[
  \mathbb{Z} \qquad \tau\tau' = \qquad (\alpha - \overline{\alpha})' = \alpha - \overline{\alpha}
\]

\[
  \begin{matrix} b, c, t \\ b', c', t' \\ b'', c'', t'' \end{matrix}
  \qquad \alpha' - \alpha = \beta\sigma
\]

\[
  A \quad \mathfrak{S}_3 \text{ y opère fidèlement}
\]

\[
  b \mapsto b' \mapsto b'' \quad c \mapsto c' \mapsto c'' \quad t \mapsto t' \mapsto t''
\]

\[
  \text{\struck{$\ill{}$}}\ \underline{\mathbb{A}}^3 \times \underline{\mathbb{A}}^3 \times \underline{\mathbb{A}}^3
\]

\[
  \mathfrak{S}_3 \times \mathfrak{S}_2 \text{ opère sur } B \qquad \text{i.e.\ } A^{\mathfrak{S}_3}
\]

\[
  \mathfrak{S}_3 \longrightarrow \mathfrak{S}_3 \times \mathfrak{S}_2 \qquad (\mathrm{id}, \mathrm{sg})
\]

\[
  \mathrm{Tr}\, u^n = P_n(\mathrm{Tr}\, u, \det u)
\]

\[
  \mathrm{Tr}\, u^{n_1} v^{m_1} u^{n_2} v^{m_2} \cdots u^{n_r} v^{m_r}
  = P_{(n_1, m_1, n_2, m_2, \ldots, n_r, m_r)}(\ill{})
\]

\[
  \lambda^n + \mu^n = P_n(\lambda + \mu, \lambda\mu)
\]

\[
  \mathrm{Tr}\, x_1 x_2 x_3 x_4 = \mathrm{Tr}\, x_1 x_2 (x_3 x_4)
\]

\[
  \text{\struck{$t_{1234}^2 - \Sigma_{1,2,(3,4)}\, t_{1234} - \cdots$}}
\]

\[
  \tau^2 - \Sigma_{1,2,(3,4)}\, \tau + \Pi_{1,2,(3,4)} = 0
\]

\[
  \Sigma_{1,2,(3,4)}, \ \Pi_{1,2,(3,4)} \in
  \text{\struck{$\mathbb{Z}[b_1, c_1, b_2, c_2$}} \ \text{\add{exprimés en} \ill{}}
\]

\[
  \underbrace{\mathrm{Tr}\, x_1}_{b_1}, \ \underbrace{\det x_1}_{c_1}, \
  \underbrace{\mathrm{Tr}\, x_2}_{b_2}, \ \underbrace{\det x_2}_{c_2}, \
  \underbrace{\mathrm{Tr}\, x_3 x_4}_{t_{3,4}}, \ \underbrace{\det(x_3 x_4)}_{c_3 c_4}
\]

\[
  \underbrace{\mathrm{Tr}\, x_1 x_2}_{t_{1,2}}, \quad
  \underbrace{\mathrm{Tr}\, x_2(x_3 x_4)}_{t_{2,3,4}}, \quad
  \underbrace{\mathrm{Tr}(x_3 x_4) x_1}_{t_{1,3,4}}
\]

\[
  \mathbb{Z}[b_1, c_1, b_2, c_2, c_3, c_4, t_{1,2}, \underbrace{t_{2,3,4}, t_{1,3,4}}_{9}]
\]

\[
  \mathbb{Z}[b_1, c_1, b_2, c_2, b_3, c_3, b_4, c_4, t_{1,2}, t_{1,3}, t_{1,4},
  t_{2,3}, t_{2,4}, t_{3,4}]
\]

\[
  \tau = \mathrm{Tr}(x_1 x_2) x_3 x_4 \qquad x_2 x_1
\]

\[
  \tau^2 - \Sigma_{(1,2),3,4}\, \tau + \Pi
\]

\[
  \mathrm{Tr}\, x_1 x_2 x_3 x_4 + \mathrm{Tr}\, x_2 x_1 x_3 x_4
\]

\[
  x_1 x_2 x_3 x_4 \quad x_2 x_3 x_4 x_1 \quad x_3 x_4 x_1 x_2 \quad x_4 x_1 x_2 x_3
\]

\[
  x_4 x_3 x_2 x_1 \quad x_3 x_2 x_1 \ldots \qquad \tau
\]

\[
  \underbrace{b_s, c_s, t_s} \ \big| \ \underbrace{t_a} \ \big| \ t_f
\]

\[
  \tau^2 \qquad \tau^8 + \alpha_1 \tau^7 + \cdots + \alpha_8 = 0
  \qquad \alpha_i \in \mathbb{Z}[b_\cdot, c_\cdot, t_{\cdot\cdot}]
\]

\[
  g\tau, \ g'\tau, \ g''\tau \qquad \textstyle\sum g\tau \qquad \sum g
\]

\[
  \tau^8 + \beta_1 \tau^7 + \cdots + \beta_8 = 0
\]

\[
  \underbrace{(\alpha_1 - \beta_1)}_{\gamma_1} \tau^7 + \cdots + \underbrace{\alpha_8 - \beta_8} = 0
\]

\[
  H \subset \mathfrak{S}_4 \qquad B^H \overset{\text{degré } 6}{\text{---}} B^{\mathfrak{S}_4} = B^{\mathfrak{S}}
\]

\[
  \alpha + \beta\tau \quad \text{à } b \text{ stables par } \mathfrak{S}_3^+
\]

\[
  \alpha' + \beta'\tau' = \alpha' + \beta'(\sigma - \tau) = (\alpha' + \beta'\sigma) - \beta'\tau
\]

\[
  \beta' = -\beta \ / \ \alpha' + \beta\sigma = \alpha
  \qquad \alpha' = \alpha + \beta\sigma \ / \ (\alpha')' = \alpha = \alpha' \ldots
\]

\[
  \begin{aligned}
    \alpha + \beta\tau + \alpha' + \beta'\tau'
    &= (\alpha + \alpha') + \bigl(\beta\tau + \beta'(\sigma - \tau)\bigr) \\
    &= \underbrace{\alpha + \alpha'} + \underbrace{\beta'\sigma} + (\beta - \beta')\tau
  \end{aligned}
\]

\[
  \alpha + \alpha' + \beta\sigma - (\alpha + \alpha' + \beta'\sigma) = (\beta - \beta')\sigma
  \qquad \text{OK}
\]

\[
  \begin{aligned}
    \tau_f + \tau_{f'} &= \Sigma_a \\
    \tau_{f'} + \tau_{f''} &= \Sigma_{a'}
  \end{aligned}
  \qquad \longrightarrow \qquad
  \tau_f - \tau_{f''} = \Sigma_a - \Sigma_{a'} = \Sigma_{\ill{}} - \Sigma_{\ill{}}
\]

\[
  \text{i.e.} \quad \Sigma_{\ill{}} + \Sigma_{\ill{}} = \Sigma_{a'} + \Sigma_{\ill{}}
\]

\[
  \Sigma_a + \Sigma_{\ill{}} = \Sigma_{a'} + \Sigma_{\ill{}}
\]

\[
  \tau_f + \tau_{f'} = \Sigma \qquad \tau_{\tilde f} + \tau_{\tilde f'} = \Sigma_{\ill{}} - \Sigma
\]

\[
  4 . 3 . 2 = 24 = 48/2 \qquad
  \tau_f \tau_{f'} = \Pi \qquad \tau_{\tilde f} \tau_{\tilde f'} = \Pi \ldots
\]

\[
  \text{donc } \{\tau_f, \tau_{f'}\} = \{\tau_{\tilde f}, \tau_{\tilde f'}\} .
\]

\[
  \tau_f - \tau_{\ill{}} = \Sigma \ldots = \Sigma_b - \Sigma_{b'}, \qquad
  \text{donc } \tau_f \neq \tau_{\tilde f} \ \text{(\uncertain{or})}, \quad \tau_f = \tau_{\tilde f'}
\]

\[
  \Sigma_{\ill{}} + \Sigma_b = \Sigma_{b'} + \Sigma_{\ill{}}
\]

\[
  \begin{aligned}
    \Sigma_{\ill{}} - \Sigma_{\ill{}} &= \Sigma_{\ill{}} - \Sigma_{\ill{}} \\
    \Sigma_{\ill{}} + \Sigma_{\ill{}} &= \Sigma_{\ill{}} - \Sigma_{\ill{}}
  \end{aligned}
\]

\[
  \Sigma_a + \Sigma_{a'} = \Sigma_b + \Sigma_{b'}
\]

\[
\Sigma_a\quad \Sigma_b\quad \Sigma_c\quad \Sigma_{AB'}\quad \Sigma_{BC'}\quad \Sigma_{CB'}\quad \Sigma_{CA'}\quad \Sigma_{AC'}
\]

\[
\Sigma_{a'}\quad \Sigma_{b'}\quad \Sigma_{c'}
\]

\[
\left\{
\begin{aligned}
\Sigma_a + \Sigma_{c'} &= \Sigma_{AB'} + \Sigma_{BC'}\\
\Sigma_a + \Sigma_{b'} &= \Sigma_{AC'} + \Sigma_{CB'}\\
\Sigma_b + \Sigma_{a'} &= \Sigma_{BC'} + \Sigma_{CA'}\\
\Sigma_b + \Sigma_{c'} &= \Sigma_{BA'} + \Sigma_{AC'}\\
\Sigma_c + \Sigma_{b'} &= \Sigma_{CA'} + \Sigma_{AB'}\\
\Sigma_c + \Sigma_{a'} &= \Sigma_{CB'} + \Sigma_{BA'}
\end{aligned}
\right.
\]

\[
\left[
\begin{aligned}
\Sigma_a + \Sigma_{CA'} &= \Sigma_b + \Sigma_{CB'}\,.\\
\Sigma_{a'} + \Sigma_{AC'} &= \Sigma_{b'} + \Sigma_{BC'}\\
\Sigma_b + \Sigma_{AB'} &= \Sigma_c + \Sigma_{AC'}\,.\\
\Sigma_{b'} + \Sigma_{BA'} &= \Sigma_{c'} + \Sigma_{CA'}\,.\\
\Sigma_c + \Sigma_{BC'} &= \Sigma_a + \Sigma_{BA'}\,.\\
\Sigma_{c'} + \Sigma_{CB'} &= \Sigma_{a'} + \Sigma_{AB'}
\end{aligned}
\right.
\]

\[
3\bigl(\Sigma_a - \Sigma_b + \Sigma_c + \Sigma_{a'} - \Sigma_{b'} + \Sigma_{c'}\bigr) + \bigl(\Sigma_{CA'} + \Sigma_{AC'} - \cdots\bigr)
\]

\[
= \bigl(\Sigma_a + \Sigma_b - \Sigma_c - \Sigma_{a'} - \Sigma_{b'} - \Sigma_{c'}\bigr) + 3\bigl(\Sigma_{CA'} + \Sigma_{AC'}\bigr)
\]

\[
(4) + (4) + (6)\qquad
\boxed{\begin{array}{ll} b_s\ \ c_s & (s\in S)\\ t_a & (a\in A)\end{array}}
\qquad\text{avec}\quad b_s = b_{\bar s},\ c_s = c_{\bar s},\ t_a = t_{\bar a}
\]

\[
b_s = \operatorname{Tr} X_s,\qquad c_s = \det X_s .
\]

\[
t_a = \operatorname{Tr} X_i X_j \;=\; \operatorname{Tr} X_j X_i \quad\text{si } \delta(a) = \{i,j\}.
\]

\[
4 + 4 + (8)\qquad
\boxed{\sigma_s,\ \pi_s,\ \mathbb{T}_s}\qquad (s\in S)
\]

\[
\mathbb{T}_s = \operatorname{Tr} X_i X_j X_k = \operatorname{Tr} X_j X_k X_i = \operatorname{Tr} X_k X_i X_j
\]

\[
\mathbb{T}_{\bar s} = \operatorname{Tr} X_k X_j X_i = \operatorname{Tr} X_j X_i X_k = \operatorname{Tr} X_i X_k X_j
\]

\[
(R_s)\ \sim\ (R_{\bar s})\qquad
\boxed{\begin{aligned}
\sigma_s &= \sigma_{\bar s} = \mathbb{T}_s + \mathbb{T}_{\bar s}\\
\pi_s &= \pi_{\bar s} = \mathbb{T}_s \cdot \mathbb{T}_{\bar s}
\end{aligned}}
\]

\begin{align*}
\sigma_s = \sigma_{\bar s} &= b_i t_{jk} + b_j t_{ki} + b_k t_{ij} - b_i b_j b_k\\
&= b_{s_1} t_{ss_1} + b_{s_2} t_{ss_2} + b_{s_3} t_{ss_3} - b_{s_1} b_{s_2} b_{s_3}
\end{align*}

\begin{align*}
\pi_s = \pi_{\bar s} &= t_{ss_1} t_{ss_2} t_{ss_3}
+ \text{\struck{$(c_{s_1} t_{ss_1} + c_{s_2} t_{ss_2} + c_{s_3} t_{ss_3})$}}\\
&\quad + \bigl[c_{s_1} t_{ss_1}(t_{s_2 s_3} - b_{s_2} b_{s_3}) + c_{s_2} t_{ss_2}(t_{s_3 s_1} - b_{s_3} b_{s_1})\\
&\qquad + c_{s_3} t_{ss_3}(t_{s_1 s_2} - b_{s_1} b_{s_2})\bigr]\\
&\quad + \bigl(b_{ss_1}^2\, c_{ss_2} c_{ss_3} + b_{ss_2}^2\, c_{ss_3} c_{ss_1} + b_{ss_3}^2\, c_{ss_1} c_{ss_2}\bigr)
\end{align*}

\[
(6)\qquad
\boxed{\begin{array}{ll} \mathbb{T}_f & (f\in F)\\ \sigma_a,\ \pi_a & (a\in A)\end{array}}
\]

\[
\text{\struck{$\mathbb{T}_f =$}}\qquad
\mathbb{T}_f = \operatorname{Tr} X_i X_j X_k X_l = \operatorname{Tr} X_j X_k X_l X_i = \operatorname{Tr} X_k X_l X_i X_j = \operatorname{Tr} X_l X_i X_j X_k
\]

\[
f = (s, t, s', t'),\quad f' = (t, s, t'', s'')
\]

\[
\boxed{\begin{aligned}
\mathbb{T}_f + \mathbb{T}_{f'} &= \sigma_a\\
\mathbb{T}_f\, \mathbb{T}_{f'} &= \pi_a
\end{aligned}}
\]

\begin{align*}
\sigma_a &= \operatorname{Tr} X_k X_l \operatorname{Tr} X_i X_j + \operatorname{Tr} X_i \operatorname{Tr} X_j X_k X_l + \operatorname{Tr} X_j \operatorname{Tr} X_k X_l X_i\\
&\qquad - \operatorname{Tr} X_i \operatorname{Tr} X_j \operatorname{Tr} X_k X_l\\
&\text{\struck{$= t_{s\ldots}\, t$}}\\
&= t_{a'a''}\, t_a + b_s \mathbb{T}_s + b_t T_t - b_s b_t t_{a',a''}
\end{align*}

\[
\overbrace{4\quad 4\quad 6}^{14}\qquad \overbrace{8\quad 6}^{14}
\]

\[
28 \text{ variables}\quad b_s,\ c_s,\ t_a,\ \mathbb{T}_s,\ \mathbb{T}_f
\]

\[
v + (c-a) = u - (d-b) + (c-a)
\]

\[
\boxed{d - b = u - v} = d - b = u - v
\]

\[
d + v = b + u,\qquad d - u = b - v
\]

\[
\alpha = a' - a
\]

\begin{align*}
S_1 &= a + b + c + d\\
S'_1 &= a + b + c + d + 4(a'-a) = -3a + b + c + d + 4a'\\
S_2 &= a + u + a' + u + (d-b) = a - b + \cdot + d + a' + 2u\\
S'_2 &= c + u + c - a + c + (a'-a) + u - d + b + c - a\\
&= -3a + b + 4c - d + a' + 2u\\
S_3 &= u + d + a' - a + u + c - a + d = -2a + \cdot + c + 2d + a' + 2u\\
S'_3 &= b + u - d + b + b + a' - a + u - d + b + c - a\\
&= -2a + 4b + c - 2d + a' + 2u
\end{align*}

\[
\begin{array}{cccccc}
a & b & c & d & a' & u
\end{array}
\]

\[
\begin{pmatrix}
1 & 1 & 1 & 1 & 0 & 0\\
-3 & 1 & 1 & 1 & 4 & 0\\
1 & -1 & 0 & 1 & 1 & 2\\
-3 & 1 & 4 & -1 & 1 & 2\\
-2 & 0 & 1 & 2 & 1 & 2\\
-2 & 4 & 1 & -2 & 1 & 2
\end{pmatrix}
\]

\[
\Delta = \det\begin{pmatrix}
1 & 1 & 1 & 4 & 0\\
-1 & 0 & 1 & 1 & 2\\
1 & 4 & -1 & 1 & 2\\
0 & 1 & 2 & 1 & 2\\
4 & 1 & -2 & 1 & 2
\end{pmatrix}
- \det\begin{pmatrix}
-3 & 1 & 1 & 4 & 0\\
1 & 0 & 1 & 1 & 2\\
-3 & 4 & -1 & 1 & 2\\
-2 & 1 & 2 & 1 & 2\\
-2 & 1 & -2 & 1 & 2
\end{pmatrix}
\]

\[
+ \det\begin{pmatrix}
-3 & 1 & 1 & 4 & 0\\
1 & -1 & 1 & 1 & 2\\
-3 & 1 & -1 & 1 & 2\\
-2 & 0 & 2 & 1 & 2\\
-2 & 4 & -2 & 1 & 2
\end{pmatrix}
- \det\begin{pmatrix}
-3 & 1 & 1 & 4 & 0\\
1 & -1 & 0 & 1 & 2\\
-3 & 1 & 4 & 1 & 2\\
-2 & 0 & 1 & 1 & 2\\
-2 & 4 & 1 & 1 & 2
\end{pmatrix}
\]

\[
\gamma' - c = a' - \beta' = c' - \beta = \alpha - a \;\Big|\; = \delta'
\]

\[
\left\{
\begin{aligned}
\gamma' &= \delta' + c\\
\beta' &= -\delta' + a'\\
\beta &= -\delta' + c'\\
\alpha &= \delta' + a
\end{aligned}
\right.
\qquad \text{reste pour compte } \underline{\gamma, \alpha'}
\]

\[
b - \gamma = \underline{\alpha - a'} = \alpha' - b' = a - \beta' \;\Big|\; = -\delta''
\]

\[
\left\{
\begin{aligned}
\gamma &= \delta'' + b\\
\alpha' &= -\delta'' + b'
\end{aligned}
\right.
\qquad
\begin{aligned}
\alpha &= -\delta'' + a'\\
\beta' &= \delta'' + a
\end{aligned}
\]

\[
\boxed{-\delta'' + a' = \delta' + a}\qquad \delta' + \delta'' = a' - a
\]

\[
\boxed{+\delta'' + a = -\delta' + a'}
\]

\[
b - \alpha' = \gamma' - c' = \gamma - b' = c - \beta = -\delta
\]

\begin{align*}
\alpha' &= \delta + b\\
\gamma' &= -\delta + c'\\
\gamma &= -\delta + b'\\
\beta &= \delta + c
\end{align*}

\[
\boxed{\text{\struck{$\delta + b = -\delta'' + b'$}}}\qquad
\boxed{-\delta + c' = \delta' + c}
\]

\[
\boxed{\text{\struck{$-\delta + b' = \delta'' + b$}}}\qquad
\boxed{\delta + c = -\delta' + c'}
\]

\[
\delta + \delta'' = b' - b,\qquad \delta + \delta' = c' - c
\]

\begin{align*}
\delta' + \delta'' &= a' - a\\
\delta'' + \delta &= b' - b\\
\delta + \delta' &= c' - c
\end{align*}

\[
\delta + \delta' + \delta'' = \tfrac12\bigl(a' + b' + c' - (a + b + c)\bigr)
= \tfrac12\bigl(\underbrace{(a'-a)}_{\varepsilon_1} + \underbrace{(b'-b)}_{\varepsilon_2} + \underbrace{(c'-c)}_{\varepsilon_2}\bigr)
\]

\[
\boxed{\left\{
\begin{aligned}
\delta &= \tfrac12(-\varepsilon_1 + \varepsilon_2 + \varepsilon_3)\\
\delta' &= \tfrac12(\varepsilon_1 - \varepsilon_2 + \varepsilon_3)\\
\delta'' &= \tfrac12(\varepsilon_1 + \varepsilon_2 - \varepsilon_3)
\end{aligned}
\right.}
\]

\begin{align*}
\alpha &= \sigma_{BC'} = \tfrac12\bigl((a'+a) \bullet (b'-b) + (c'-c)\bigr)\\
\beta &= \sigma_{AB'} = \tfrac12\bigl(-(a' \bullet a) + (b'-b) + (c'+c)\bigr)\\
\gamma &= \sigma_{CA'} = \tfrac12\bigl((a'-a) + (b'+b) - (c'-c)\bigr)\\
\alpha' &= \sigma_{AC'} = \tfrac12\bigl(-(a'-a) + (b'+b) + (c'-c)\bigr)\\
\beta' &= \sigma_{CB'} = \tfrac12\bigl((a'+a) + (b'-b) - (c'-c)\bigr)\\
\gamma' &= \sigma_{BA'} = \tfrac12\bigl((a'-a) \bullet (b'-b) + (c'+c)\bigr)
\end{align*}

\[
-c' + c + 2c',\qquad -b' + b,\qquad -a' + a,\qquad c' - c
\]

\begin{align*}
\varphi_a &= (a'-a) + 2(b+c)\\
\varphi_b &= (b'-b) + 2(a+c)\\
\varphi_c &= (c'-c) + 2(a+b)
\end{align*}

\begin{align*}
\varphi_{a'} &= (b'+c') \bullet (a'-a) + b' + c' = 2(b'+c') - (a'-a)\\
\varphi_{b'} &= a' + c' + a' - (b'-b) + c' = 2(c'+a') - (b'-b)\\
\varphi_{c'} &= (a'+b') + a' + b' - (c'-c) = 2(a'+b') - (c'-c)
\end{align*}

\[
\begin{array}{c}
\begin{array}{cccccc} a & b & c & a' & b' & c' \end{array}\\
\det\left(\begin{array}{ccc|ccc}
-1 & 2 & 2 & 1 & 0 & 0\\
2 & -1 & 2 & 0 & 1 & 0\\
2 & 2 & -1 & 0 & 0 & 1\\
\hline
1 & 0 & 0 & -1 & 2 & 2\\
0 & 1 & 0 & 2 & -1 & 2\\
0 & 0 & 1 & 2 & 2 & -1
\end{array}\right)
\end{array}
= \det\begin{pmatrix} 27 & 1\\ 1 & 27\end{pmatrix}
= 27^2 - 1 \in \mathbb{Q}^*
\]

\[
= (3^3)^2 - 1 = 3^6 - 1 = 728 = 8\cdot 91 = \boxed{2^3\cdot 7\cdot 13}
\]

\[
-1 + 8 + 8 - (-4 - 4 - 4) = 16 + 12 - 1 = 27
\]

\[
\begin{array}{r}
27\\
27\\
\hline
189\\
54\phantom{0}\\
\hline
729
\end{array}
\]

\begin{align*}
A &= a + a' = \lambda\\
B &= b + b' = \mu\\
C &= c + c' = \nu
\end{align*}

\begin{align*}
A' &= a' + a\\
B' &= b' + b\\
C' &= c' + c
\end{align*}

\begin{align*}
a + a' &= \lambda\\
b + b' &= \mu\\
c + c' &= \nu
\end{align*}

\[
\varphi_a + \varphi_{a'} = 2(b + b' + c + c') = 2(\mu + \nu)
\]