I have two operators $A$ and $B$ such that
$[A,B] = C$
$[A,C] = -2A$
$[B,C] = +2B$
and I would like to obtain an expression for $\log(\exp(A+B)\exp(-B)\exp(-A))$. Is it a linear combination of $A$, $B$ and $C$? If yes, is there a way to calculate explicitly the coefficients? (I had a look at generalizations of the CBH formula, but this was not very illuminating regarding the second part of my question, i.e. how to calculate the coefficients)