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 consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. We have established that $\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$. By applying the induction hypothesis, show that the generic nilindex of $\mathcal{V}$ is greater than or equal to $n-1$, and that if moreover $\mathcal{V} x = \{0\}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.
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 consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. We have established that $\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
By applying the induction hypothesis, show that the generic nilindex of $\mathcal{V}$ is greater than or equal to $n-1$, and that if moreover $\mathcal{V} x = \{0\}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.