Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. Let $G$ be the set of elements $u \in E$ satisfying: (a) $u \in \operatorname{Im}(T)$, (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$. Deduce that the map $(v,w) \in G \times G \mapsto (T(v) \mid T(w))$ is a scalar product on $G$.
Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T$ be an endomorphism of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$. Let $G$ be the set of elements $u \in E$ satisfying:
(a) $u \in \operatorname{Im}(T)$,
(b) $\forall v \in E, S(u,v) = 0$,
where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Deduce that the map $(v,w) \in G \times G \mapsto (T(v) \mid T(w))$ is a scalar product on $G$.