Let $A\in M_n(\mathbb{C})$ is nilpotent and $B\in M_n(\mathbb{C})$. If $AB-BA$ and $A$ commute, prove that $AB$ is nilpotent.
If you can prove that the Lie algebra generated by $A$ and $B$ is solvable, the problem will be finished by Lie theroem. But I can't prove it.
