I am reading Nielsen and Chuang on P. 291, for anyone interested in the origin of my question.
Given an annihilator $a$ and its corresponding creator $a^\dagger$ such that $[a,a^\dagger] = 1$ and another annihilator $b$ with creator $b^\dagger$, an argument in a proof claims the following:
Let $G = a^\dagger b - ab^\dagger$. Then, $[G,a] = -b$ and $[G,b] = a$.
I don't see how these two relations hold. Can someone please point me in the right direction or prove them?
Thank you SOCommunity!