Consider the following exponential of matrices $\exp(X+\delta Y)$, where $\delta$ is a smaller number, and $X,Y$ are non-commuting matrices. I am interested in expanding it in such a way that
$$
\exp(X+\delta Y) = e^Xe^{\delta Y}e^{\delta A\_1}e^{\delta^2 A\_2}e^{\delta^3 A\_3}...,
$$
namely that organizing the terms of the expansion according to the order of $\delta$.


From [wikipedia](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#:%7E:text=this%20Hopf%20algebra.-,Zassenhaus%20formula,-%5Bedit%5D), we have
$$
e^{t(X+Y)} = e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~
e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~
e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \cdots
$$
hence it is naturally to guess that
$$
A\_1 = -\frac{1}{2}[X,Y]+\frac{1}{6}[X,[X,Y]]... = \sum\_{n=2}^\infty\frac{(-1)^{n+1}}{n!}[X,[X,[X,...[X,Y]].
$$


This seems to be correct for the particular example I have in mind, but I don't have proof for that. Also I couldn't figure out the higher order terms $A\_{2,3,...}$. Any help is really appreciated.



Reorganizing the terms in the Baker–Campbell–Hausdorff formula (or Zassenhaus formula) for $\exp(X+\delta Y)$ for small $\delta$
This expansion is derived by K. Kumar in [On Expanding the Exponential](https://doi.org/10.1063/1.1704742), see equation (9) (with $t=1$) and section 6.


