Validity of the Baker-Campbell-Hausdorff formula when convergent

Question

I have found that the Baker-Campbell-Hausdorff formula

$\exp(X)\exp(Y)=\exp(X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]+\frac{1}{12}[Y,[Y,X]]+\dotsc)$

is only valid for sufficiently small $X$ and $Y$, otherwise the infinite sum of commutators may not be convergent (by for example wikipedia and B. Hall).

I was not able to find out whether the formula is guaranteed to be correct whenever it is convergent. Does anyone know or know where to look for an answer to this?

Edit: In particular I would like to use this formula for matrices of which I know that big enough commutators on the right-hand side are eventually zero.

Answer 1

To address the edit:

equality holds whenever the Lie subalgebra generated by $\{X,Y\}$ is nilpotent, by a variation of the argument given by Andreas Cap:

in this case, use analyticity of the functions given by $t\mapsto\exp(tX)\exp(tY)$ and the function $\exp(B(tX,tY))$ where $B$ is the BCH expansion (which is a finite sum in this case). Both functions are real analytic on $\mathbf{R}$ and coincide around zero. Hence they are equal.

Answer 2

I think this should at least be true on any connected neighborhood of $0$ on which the series converges by analyticity: Lie groups are automatically anaytic manifolds, and both multiplication and the exponential map are analytic. Hence both sides of the formula represent analytic functions, with the left hand side defined on $\mathfrak g\times\mathfrak g$ and an the right hand side defines whehere the power series converges. Since these functions agree locally around the origin, uniqueness of analytic continuation shows that they agree on any connected neighborhood of the identity, on which the series converges. I don't know whether there may be other domains in which the series converges and what happens there.