In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. By using $f_{4}$, show that $f_{3}$ cannot be equal to $\pm f_{1}f_{2}$.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Show that $f_{1}f_{2}f_{3}$ is an orthogonal automorphism, symmetric and not collinear to $\operatorname{Id}_{E}$.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. What is the spectrum of $f_{1}f_{2}f_{3}$? Show that there exists $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. We fix such an $x$ for the rest.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We fix $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. Show that $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$ is an orthonormal family.

In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We set $V = \operatorname{Vect}(F)$ where $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$. It is therefore a vector subspace of $E$ of dimension 8. a) Show that $V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}$. b) We denote by $f_{i}^{\prime}$ the endomorphism induced by $f_{i}$ on $V^{\perp}$, $i = 1, 2, 3$. Justify that there exists $\delta^{\prime} \in \{-1, 1\}$ such that $f_{3}^{\prime} = \delta^{\prime} f_{1}^{\prime} f_{2}^{\prime}$. If necessary, by replacing $f_{3}$ by $-f_{3}$, we consider for the rest that $f_{3}^{\prime} = f_{1}^{\prime} f_{2}^{\prime}$. c) Let $e$ be fixed in $V^{\perp}$, of norm 1. By proceeding as in II.B.1.a) (but this is not to be redone), one can show that $(e, f_{1}(e), f_{2}(e), f_{1}f_{2}(e))$ is an orthonormal basis of $V^{\perp}$. By noting that $f_{3}(e) = f_{1}f_{2}(e)$, use this basis to show that: $\forall y \in V^{\perp}, f_{4}(y) \in V$. Thus $W = f_{4}(V^{\perp})$ is a vector subspace of $V$ of dimension 4. d) Show that the sum of $W$ and $V^{\perp}$ is direct and that $W \oplus V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}, f_{4}$. Then reach a contradiction.

In this section, the dimension of $E$ is 12. Deduce the value of $d_{12}$.

In this section, the dimension of $E$ is 8. Show that, for all $(x_{0}, \ldots, x_{7}) \in \mathbb{R}^{8}$, $$\left(\begin{array}{cccccccc} x_{0} & -x_{1} & -x_{2} & -x_{4} & -x_{3} & -x_{5} & -x_{6} & -x_{7} \\ x_{1} & x_{0} & -x_{4} & x_{2} & -x_{5} & x_{3} & -x_{7} & x_{6} \\ x_{2} & x_{4} & x_{0} & -x_{1} & -x_{6} & x_{7} & x_{3} & -x_{5} \\ x_{4} & -x_{2} & x_{1} & x_{0} & x_{7} & x_{6} & -x_{5} & -x_{3} \\ x_{3} & x_{5} & x_{6} & -x_{7} & x_{0} & -x_{1} & -x_{2} & x_{4} \\ x_{5} & -x_{3} & -x_{7} & -x_{6} & x_{1} & x_{0} & x_{4} & x_{2} \\ x_{6} & x_{7} & -x_{3} & x_{5} & x_{2} & -x_{4} & x_{0} & -x_{1} \\ x_{7} & -x_{6} & x_{5} & x_{3} & -x_{4} & -x_{2} & x_{1} & x_{0} \end{array}\right)$$ is a similarity matrix. What can we deduce from this?

Conjecture the value of $d_{n}$ in the general case.

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') = 1$. Let $x \in F$ with $x \neq 0$. Set $y = f(x)$.
a) Show that $q(x+y)$ or $q(x-y)$ is non-zero.
b) Prove Witt's theorem in this case, using question III.A.2-d).

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
We now assume that $F$ and $F'$ are non-singular, with $\operatorname{dim}(F) = \operatorname{dim}(F') > 1$.
a) Show that there exist $F_1$ and $F_2$ non-singular, such that $F_1 \perp F_2$ and $F = F_1 \oplus F_2$, with $\operatorname{dim}(F_1) = \operatorname{dim}(F) - 1$.
b) Suppose that there exists $g \in O(E,q)$ such that $g_{/F_1} = f_{/F_1}$. Denote $F_1' = f(F_1)$. Show that $f(F_2) \subset F_1'^\perp$ and that $g(F_2) \subset F_1'^\perp$.
c) Show that there exists $$h \in O\left(F_1'^\perp, q_{/F_1'^\perp}\right) \text{ such that } h_{/g(F_2)} = (f \circ g^{-1})_{/g(F_2)}.$$
d) Show that there exists $k \in O(E,q)$ such that $k_{/F} = f$.

We propose to prove Witt's theorem, whose statement is: ``let $F$ and $F'$ be two vector subspaces of $E$ such that there exists an isometry $f$ from $(F, q_{/F})$ to $(F', q_{/F'})$. Then there exists $g \in O(E,q)$ such that $g_{/F} = f$.''
Prove Witt's theorem.

Let $\alpha$ be a non-zero element of $E$. Show, for every vector $x$ of $E$, the identity: $$\tau _ { \alpha } ( x ) = x - 2 \frac { \langle \alpha , x \rangle } { \langle \alpha , \alpha \rangle } \alpha$$

We assume in this question that the space $E$ has dimension 1. Show that the root systems of $E$ are the sets $\{ \alpha , - \alpha \}$, with $\alpha \in E \backslash \{ 0 \}$.

In this question, the space $E$ has dimension $n \geq 2$. For every pair $(\alpha , \beta)$ of non-zero vectors of $E$, let $\theta _ { \alpha , \beta }$ be the geometric angle between $\alpha$ and $\beta$, that is, the unique element of $[ 0 , \pi ]$ given by: $\| \alpha \| . \| \beta \| \cos \theta _ { \alpha , \beta } = \langle \alpha , \beta \rangle$.
Let $\mathcal { R }$ be a root system of $E$ and let $\alpha , \beta$ be two non-collinear elements of $\mathcal { R }$.
a) Show, using property 4, that: $2 \frac { \| \alpha \| } { \| \beta \| } \left| \cos \theta _ { \alpha , \beta } \right| .2 \frac { \| \beta \| } { \| \alpha \| } \left| \cos \theta _ { \alpha , \beta } \right| \leq 3$.
b) Assume $\| \alpha \| \leq \| \beta \|$. Show that the pair $(\alpha , \beta)$ is found in one of the configurations listed in the table below (each row corresponding to a configuration):

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

In this question, the space $E$ has dimension $n \geq 2$. Conversely, assume that a pair $(\alpha , \beta)$ of non-collinear vectors of $E$ is found in one of the configurations listed in the table below. Show that the real number $2 \frac { \langle \alpha , \beta \rangle } { \langle \alpha , \alpha \rangle }$ is an integer; specify its value.

$\theta _ { \alpha , \beta }$	$\cos \theta _ { \alpha , \beta }$	$\| \beta \| / \| \alpha \|$
$\pi / 2$	0	$\geq 1$
$\pi / 3$	$1 / 2$	1
$2 \pi / 3$	$- 1 / 2$	1
$\pi / 4$	$\sqrt { 2 } / 2$	$\sqrt { 2 }$
$3 \pi / 4$	$- \sqrt { 2 } / 2$	$\sqrt { 2 }$
$\pi / 6$	$\sqrt { 3 } / 2$	$\sqrt { 3 }$
$5 \pi / 6$	$- \sqrt { 3 } / 2$	$\sqrt { 3 }$

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ Show that $\theta _ { \mathcal { R } }$ is well-defined and equals $\pi / 2 , \pi / 3 , \pi / 4$ or $\pi / 6$.

In this question, the space $E$ has dimension $n = 2$. For every root system $\mathcal { R }$ of $E$, we set $$\theta _ { \mathcal { R } } = \min \left\{ \theta _ { \alpha , \beta } \mid ( \alpha , \beta ) \in \mathcal { R } ^ { 2 } , \alpha \neq \beta \text { and } \alpha \neq - \beta \right\}$$ For each value of $k \in \{ 2,3,4,6 \}$, draw graphically a root system $\mathcal { R } _ { k }$ such that $\theta _ { \mathcal { R } _ { k } } = \pi / k$. It is not necessary to justify that the figures drawn represent root systems. What is the cardinality of $\mathcal { R } _ { k }$? No justification is required.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Show that the vector subspace of $E$ spanned by the set $\mathcal { R } _ { 0 }$ is a vector plane.

In this question, the space $E$ has dimension $n = 3$. Let $(e _ { 1 } , e _ { 2 } , e _ { 3 })$ be an orthonormal basis of $E$ and $\mathcal { R } _ { 0 } = \left\{ e _ { i } - e _ { j } \mid 1 \leq i , j \leq 3 , i \neq j \right\}$.
Draw graphically $\mathcal { R } _ { 0 }$ in the plane $\operatorname { Vect } \left( \mathcal { R } _ { 0 } \right)$. Recognize one of the root systems represented in question I.D.2.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
Show that $\mathcal { A }$ is a vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket. Show that we have $\mathcal { A } _ { 0 } = \mathcal { E }$, where $\mathcal { A } _ { 0 }$ denotes $\mathcal { A } _ { \lambda }$ when $\lambda$ is the zero linear form. Give a basis of $\mathcal { A } _ { 0 }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
For $k \in \{ 1,2 \}$, we denote by $e _ { k }$ the element of $\mathcal { E } ^ { * }$ which associates to every matrix $\left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right)$, where $D = \left( \begin{array} { c c } d _ { 1 } & 0 \\ 0 & d _ { 2 } \end{array} \right) \in \mathcal { D } ( 2 , \mathbb { R } )$, the coefficient $d _ { k }$.
a) Verify that $(e _ { 1 } , e _ { 2 })$ forms a basis of $\mathcal { E } ^ { * }$.
We equip $\mathcal { E } ^ { * }$ with the unique inner product making $(e _ { 1 } , e _ { 2 })$ an orthonormal basis.
b) Let $\mathcal { R } = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$. Show that the set $\mathcal { R }$ is a root system of $\mathcal { E } ^ { * }$. For this, you may draw the set $\mathcal { R }$ in the Euclidean space $\mathcal { E } ^ { * }$ and recognise one of the root systems encountered in question I.D.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Let $\alpha \in \mathcal { R }$. Determine by calculation the vector subspace $\mathcal { A } _ { \alpha }$. Verify that $\mathcal { A } _ { \alpha }$ is a one-dimensional vector space.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Establish the relation $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Prove the equality $\mathcal { S } ( \mathcal { A } ) = \mathcal { R }$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
We now set $\alpha = e _ { 1 } - e _ { 2 }$, $\beta = 2 e _ { 2 }$, $H _ { \alpha } = \left( \begin{array} { c c c c } 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$ and $H _ { \beta } = \left( \begin{array} { c c c c } 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array} \right)$.
a) Using the results from question III.C.3, show that there exists a pair $\left( X _ { \alpha } , X _ { - \alpha } \right) \in \mathcal { A } _ { \alpha } \times \mathcal { A } _ { - \alpha }$ and a pair $\left( X _ { \beta } , X _ { - \beta } \right) \in \mathcal { A } _ { \beta } \times \mathcal { A } _ { - \beta }$ such that $( X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } )$ and $( X _ { \beta } , H _ { \beta } , X _ { - \beta } )$ are admissible triples of $\mathcal { A }$.
b) Show that $\mathcal { A }$ is the smallest vector subspace of $\mathcal { M } ( 4 , \mathbb { R } )$ stable by bracket and containing the matrices $X _ { \alpha } , H _ { \alpha } , X _ { - \alpha } , X _ { \beta } , H _ { \beta }$ and $X _ { - \beta }$.