I am interested in evaluating the commutator $[e^V,e^W]$ where we know that $[V,W]=c$, and $V$ and $W$ are vector fields.
I know that $e^V= 1 + V + \frac{V^2}{2}+\frac{V^3}{6}+\dots$, so I can see that, say, $[V, e^W]=[V,1 + W + \frac{W^2}{2}+\frac{W^3}{6}+\dots]=[V,1]+[V,W]+\frac{1}{2}[V,W^2]+\dots$
or something along those lines (which is apprently supposed to simplify to $ce^W$?), but trying to expand $[e^V, e^W]$ seems impossible to do reasonably, yet the answer is supposed to be fairly simple, at least when put to a low order. Is there any tricks or obvious thing I'm missing?
Well, if the commutator trivializes to a constant, $[V,W]=c$, the CBH expansion collapses to the trivial form, $$ e^V e^W= e^{V+W+ c/2}, $$ that is the expansion terminates after the leading term.
You then have $$[e^V,e^W]= e^{V+W+ c/2} - e^{V+W- c/2}= e^{V+W} ~2\sinh (c/2). $$
A word of warning: taking Lie algebra commutators of group elements, instead of group commutators, is highly peculiar, and you must double-check your application.