Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric. a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$. b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$. We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product. c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?
Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric.\\
a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$.\\
b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$.\\
We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product.\\
c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?