If complex matrices $A$, $B$, $AB-BA$ are nilpotent, show that $A+B$ is nilpotent.

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.

Mathematics Asked by Rex Wang on November 12, 2021

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