I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.

Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$

$$ \log(e^X e^Y) = X + \frac{\ad_X}{1 - e^{-\ad_X}}Y + O(Y^2),$$

but it doesn't provide a reference for this formula. Who discovered this representation of the BCH formula and does it hold in the general case for general $X$ and $Y$ or is this only valid in the case $[X,Y] = sY$?

Another question related to this, is there a name for when $[X,Y] = sY$?

A third question is, what is the proper formal Taylor expansion for the $\frac{\ad_X}{1-e^{-\ad_X}}$ term? Is this convergence of this formal Taylor expansion limited by any particular spectral radius? The wiki article does not seem to go into detail about that.