I'm trying to find the condition necessary for this commutator relationship equality:
$[A,B^2]=2B[A,B]$
So far I've done this: \begin{align*} [A,B^2] & = B[A,B] + [A,B]B \\ & = BAB - BBA + ABB - BAB. \end{align*} Now, its apparent that $B[A,B]$ must equal $[A,B]B$ which is true if $B$ commutes with $[A,B]$. If that's true, what can I say about $[A,B]$? I've played around with identity $BB^{-1}$ and adding things like $AAB - AAB$ but I really can't make that step forward to massage my equation to equal $B[A,B] + B[A,B]$ or $BBB^{-1}[A,B] + BBB^{-1}[A,B]$ Can anyone help me make this more clear or is the answer simply just $B$ must commute with $[A,B]$ so $[A,B]B = B[A,B]$? Thanks.