In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1. a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that: $$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$ Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$. b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$. c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.
In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1.\\
a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that:
$$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$
Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$.\\
b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$.\\
c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.