linear algebra: one more case, when $ A + B $ is a nilpotent matrix, when both $ A $ and $ B $ are nilpotent

How to prove that $ A + B $ is nilpotent, when $ A $, $ B $, $[A, B]$ they are nilpotent matrices, and also $ A $ Y $[A, B]$, $ B $ Y $[A, B]$ They are pairs of interchangeable matrices.

It seems that you should use a binomial formula for interchangeable matrices, but if
$$ A * [A, B] = [A, B] * A \ B * [A, B] = [A, B] * B $$ it means that $ A $ Y $ B $ Is it interchangeable?