QII.C.5
Groups
Symplectic and Orthogonal Group Properties
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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.