I wonder how many different ways are there of writing the Baker-Hausdorff equation! This is a form which I recently encountered and haven't been able to figure out how it comes,
$e^ae^Xe^b = e^{[X+L_X.(b-a)+(L_X \coth L_X).(b+a)+..]}$
where $L_X$ is an operator which acts on another matrix say $f$ as,
$L_X.f = \frac{1}{2}[X,f]$
I would be grateful if someone can help with the above.
(and motivate it)