Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $A$ and $B$. However, this formula holds only for sufficiently small $A$ and $B$.
My question is for general anti-Hermitian operators $A$ and $B$, which are not necessarily small, is it still true that there exists an operator $C$ in the Lie algebra generated by $A$ and $B$ such that $e^C=e^A e^B$?