Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ 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 $x \in E$ of norm 1. a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$. b) Deduce that $d_{n-4} \geqslant 3$.
Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ 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 $x \in E$ of norm 1.\\
a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$.\\
b) Deduce that $d_{n-4} \geqslant 3$.