Let $A$ and $B$ be two matrices.
I am trying to prove the following formula (and also find the conditions on $A$ and $B$ for it to work, if it is not true for any $A$ and $B$) :
$$\underset{n=0}{\sum^j} \binom{j+1}{n}\text{Tr}\left(\left[(A)^n, B\right]\left[(A)^{j-n}, B\right]\right) = \text{Tr}\left(B{\left[(A)^{j}, B\right]}\right)$$
$\left[(A)^n, B\right]$ is the iterated application of the adjoint of $A$, aka the iterated commutator $m$ times: $$(ad_X)^{(n)}(Y) = [(X)^n,Y] \equiv \underbrace{[X,\dotsb[X,[X}_{n \text {times }}, Y]] \dotsb],\quad [(X)^0,Y] \equiv Y $$.
I must admit I am a bit stumped, I only managed to get to:
$$ \underset{n=0}{\sum^j} \binom{j}{n}\left[(A)^n, B\right]\left[(A)^{j-n}, B\right] = \left[(A)^{j}, B^2\right]$$
I would be grateful for any help one might provide,
Thank you in advance!