= Nilpotence of commutation by a nilpotent endomorphism
{title2=$N^q=0\Longrightarrow(\operatorname{ad}N)^{2q-1}=0$}
For a <nilpotent endomorphism> $N$ of a <vector space>, the <commutator> action $\operatorname{ad}N:T\mapsto NT-TN$ on its <endomorphisms> is nilpotent. The commuting left and right multiplication maps give $(\operatorname{ad}N)^r(T)=\sum_{j=0}^r(-1)^j\binom rjN^{r-j}TN^j$. If $N^q=0$ and $r\geq2q-1$, every summand vanishes. This works in every <characteristic of a field> and is useful when applying <Engel theorem> to an <Adjoint representation>.
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