I am looking for a closed-form of the second-order term in the Baker–Campbell–Hausdorff (BCH) expansion of $\log{e^X e^Y}$. Specifically, in this Wiki article, one has
$$ \log{e^X e^Y} = X + \frac{\mathrm{ad}_X}{I - e^{-\mathrm{ad}_X}} Y + O (Y^2) . \tag{1} $$
I would like to know the explicit form of the operator $U(X)$ that would go with the $O(Y^2)$-term in (1). For example, something like
$$ \log{e^X e^Y} = X + \frac{\mathrm{ad}_X}{I - e^{-\mathrm{ad}_X}} Y + Y U(X) Y + O(Y^3) , \tag{2} $$
or similar.
Remarks. I understand that the linear term $\frac{\mathrm{ad}_X}{I - e^{-\mathrm{ad}_X}}$ in (1) follows from the integral formula here via first-order Taylor expansion with respect to $Y$ about $Y = 0$. Hence, the second-order term I seek sould follow from retaining the quadratic order in the Taylor expansion.