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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