Conversely, let $(h_{1}, \ldots, h_{d-1})$ be a family of $\mathscr{L}(E)$ such that the $h_{i}$ are antisymmetric orthogonal automorphisms satisfying for all $i \neq j$, $h_{i}h_{j} + h_{j}h_{i} = 0$. Show that $\operatorname{Vect}\left(\operatorname{Id}_{E}, h_{1}, \ldots, h_{d-1}\right)$ is a vector subspace of $\mathscr{L}(E)$, of dimension $d$, included in $\operatorname{Sim}(E)$.