Looking for a proof regarding Baker-Campell-Hausdorff

[Kontilera]

Hello!
Is there any nice proof that if [A,B] belongs to the same vectorspace as A and B, then C is in the same vectorspace to all orders, given that
e^Ae^B = e^C
?
It is obvious to the second order but at higher orders it seems as if terms will cancel but I can't prove it.

Thanks!

 

[fzero]

Yes, it's actually not enough for $C$ to be in the same vector space. The quantities $A,B,\[A,B\]$ and higher commutators actually close as a Lie algebra. This is still a vector space, but might be much smaller than the original vector space. $C$ will be in this Lie algebra. I don't have a reference for a detailed proof at hand, but the wiki discussion has an outline based on what is called Friedrichs' theorem.