Jacobson lemma for a commuting commutator
= Jacobson lemma for a commuting commutator
{c}
Let $C=[B,A]$ and suppose $AC=CA$. Then $C$ is nilpotent. For the derivation $D(X)=[B,X]$, one has $D(p(A))=p'(A)C$. If $f(A)=0$, induction gives
$$
f^{(k)}(A)C^{2^k-1}=0.
$$
Indeed, if $uC^m=C^mu=0$, differentiating $uC^m=0$ and multiplying on the left by $C^m$ gives $C^mD(u)C^m=0$; for $u=f^{(k)}(A)$ this replaces $m$ by $2m+1$. Taking $k=\deg f$ makes $f^{(k)}$ a nonzero constant and proves nilpotence.