Note on Wermuth's theorem on commuting operator exponentials

We apply Wermuth’s theorem on commuting operator exponentials to show that if $A,B\in B(X)$, $X$ being Banach space and $A$ of $2\pi i$-congruence free spectrum, then $e^{A}B=B{e}^A$ if and only if $AB=BA$. We employ this observation to provide alternative proof of similar result by Chaban and Mortad, applicable for $X$ being a Hilbert space.