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 established that $K(\mathcal{V}) = \operatorname{Vect}(\mathcal{V}^{\bullet}) \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$ (from question 23 applied to all $v \in \mathcal{V}$), and Lemma B states that if $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$ then $v(x) = 0$ for every $v \in \mathcal{V}$.
Conclude the proof of Gerstenhaber's theorem: if $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.