We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$, and given $x \in E$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$.
Lemma B states: Let $x$ be in $\mathcal{V}^{\bullet} \backslash \{0\}$. If $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$, then $v(x) = 0$ for every $v$ in $\mathcal{V}$.
Let $x \in \mathcal{V}^{\bullet} \backslash \{0\}$ such that $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$. We choose $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$.
Given $y \in K(\mathcal{V})$, show that for every $k \in \mathbf{N}$ there exist $y_{k} \in K(\mathcal{V})$ and $\lambda_{k} \in \mathbf{R}$ such that $y = \lambda_{k} x + u^{k}(y_{k})$. Deduce that $K(\mathcal{V}) \subset \operatorname{Vect}(x)$ and then that $v(x) = 0$ for every $v \in \mathcal{V}$.