From what I read, $BCH$ formula states that, when $X,Y\in\mathfrak g$ are small enough, then we have $\exp(X)\exp(Y)=\exp(Z)$ where
$$Z=(X+Y)+\frac{1}{2}[X,Y]+\frac{1}{12}\left([X,[X,Y]]-[Y,[X,Y]]\right)+\cdots$$
My question is: Suppose $\exp(X)\exp(Y)=\exp(Z)$ for some $X,Y,Z$ with no requirement on how small $X,Y$ are. Does the $BCH$ formula still hold?