Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements

$$ G = \langle{e^{tX},e^{sY}\rangle}$$ for all $t,s$. The Lie product formula tells us that $e^{u(X+Y)} \in G$. What about the commutator? E.g., for all $u$, what conditions are necessary so that $$ e^{u [X,Y]} \in G$$ Clearly we can achieve this for ``infinitesimal'' $u$ by considering something like $e^{t X} e^{t Y} e^{-t X} e^{-t Y}$ for infinitesimal $t$. But what about for finite $u$? Is there a formula like $$ e^{u [X,Y]} = \mathcal{P} e^{\int X t' + Y s' d\tau }$$ for some $t(\tau), s(\tau)$? If so, is there an explicit formula for $t$ and $s$ in terms of $u$? I know of such a formula for SU(2), but I do not know how general group.