Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]).$$
As I am not an expert in Lie algebras, I was wondering if this setup is suitable to say that thus a formula like this holds when $X,Y$ are unbounded operators with common core such that $[X,Y]=i$. If so, I would be grateful for the reference to a proof in that particular case.
Motivation: An important identity in quantum mechanics is the decomposition of the Weyl operator as an exponential of position and momentum operators: $$W(z_1,...,z_n)=\exp(-i\sqrt{2}\sum_{j}(x_jp_j-y_jq_j)).$$ When I was trying to prove this in an explicit way (straight from the definition of the Weyl operator), there comes a point when I need to use the BCH formula. Since for unbounded operators I cannot use the usual expressions for exponential (as a sum of a convergent series), this worries me slightly.