Lemma 1

Let $V$ be a vector space over $\mathbf{K}$. If $x$ is a nilpotent endomorphism of $V$, the mapping $y \mapsto [x, y]$ of $\mathscr{L}(V)$ into $\mathscr{L}(V)$ is nilpotent. If $f$ denotes this mapping, $f^m(y)$ is a sum of terms of the form $\pm x^i y x^j$ with $i + j = m$. If $x^k = 0$, then $f^{2k-1}(y) = 0$ for all $y$.