If $\Pi:G\to GL(V)$ is a representation (probably, with additional conditions wich I don't know) of $G$ a Matrix Lie group and $\text{GL}(V)$ is the invertible linear operators from $V$ to $V$ with $V$ a vector space, then, $e^{X}e^{Y}=e^{Z(X,Y)}$ implies that $e^{\Pi(X),\Pi(Y)}=e^{\Pi(Z(X,Y))}$?. In other words, If Baker-Cambell-Hausdorff formule is valid in $G$, then is valid on $GL(V)$?
I ask this question because I am studying the Schrodinger representation of the Heisenberg group.
Thanks.