Let $\mathfrak{g}$ be a Lie algebra of some Lie group $G$. Suppose that $\mathfrak{g}$ is equipped with some norm $\|\cdot\|$, so that for some $K>0$ we have $\|[X,Y]\|\leq K\|X\|\|Y\|$ for all $X,Y\in\mathfrak{g}$.
I would like to have some general and `useful' norm estimate on $\|\log\big(\exp(X)\exp(Y)\big)\|$, assuming that $\|X\|$ and $\|Y\|$ are sufficiently small, which is expressed in terms of $K$, $\|X\|$ and $\|Y\|$. This should be possible by applying the Baker-Campbell-Hausdorff formula to $\log\big(\exp(X)\exp(Y)\big)$, and it looks like it must have been done before somewhere. Is there any reference?