We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$. Suppose that there exists $v_{0}$ in $\mathcal{V}$ such that $v_{0}(x) \neq 0$. Let $v \in \mathcal{V}$. By considering $v + tv_{0}$ for $t$ real, show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$.
We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Suppose that there exists $v_{0}$ in $\mathcal{V}$ such that $v_{0}(x) \neq 0$. Let $v \in \mathcal{V}$. By considering $v + tv_{0}$ for $t$ real, show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$.