If nilpotent matrix $A$ and $AB−BA$ commute,show that $AB$ is nilpotent.

Let $$A$$ and $$B$$ be $$n×n$$ complex matrices.

If $$A$$ is an nilpotent matrix, and $$A$$ commute with $$AB−BA$$ , show that $$AB$$ is nilpotent.

Equivalently, the question can be expressed as following description.

Let $$A$$ and $$B$$ be $$n×n$$ complex matrices.

Definite linear transformation $$T$$ as $$T(B)=AB-BA$$.

If $$A$$ is an nilpotent matrix, and $$T^2(B)=0$$ , show that $$AB$$ is nilpotent.

I’ve known that $$AB-BA$$ is nilpotent.

Furtherly, if $$A^m=0$$ , by considering $$T^n(B)=\sum_{i=0}^n(-1)^iA^{n-i}BA^i$$ , i found that $$A^kBA^l=0$$ when $$k+l\geqslant m$$.

But I don’t know how to continue, thanks for any help.