 Jacobson lemma for a commuting commutator

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