Let $A$ and $B$ be square complex matrices, and $C=[A,B]=AB-BA$. Suppose that $A$, $B$, and $C$ are nilpotent. When $A$ and $C$ commute, as well as $B$ and $C$ commute, then $A+B$ is nilpotent. The answer can be found here.
I wonder if this is true in general. That is, is the following statement true?
If complex square matrices $A$, $B$, $AB-BA$ are nilpotent, then $A+B$ is nilpotent.
This question occurs when I was studying Lie algebras.
If $~\text{ad}x, \text{ad}y~$, and $~\text{ad}[x,y]$ are nilpotent, so is $~\text{ad}(x+y)$?
I have tried some simple types, but can't find a counter-example.
Can anyone prove that this is true or just give a counter-example? Thanks a lot.