As I understand it, for two non-commuting operators $X$ and $Y$, their exponentiated sum $e^{X+Y}$ can be expanded, via the so-called Zassenhaus formula, as follows $$e^{t(X+Y)} = e^{tX}\,e^{tY}\,\prod_{n=2}^{\infty}e^{t^nC_n}\; , \qquad (1)$$ where, e.g., $C_2=[X,Y]$ and $C_3=\frac{1}{3}[C_2,X+2B]$, and in general, $$C_n = \frac{1}{n!}\Big[\frac{\partial^n}{\partial t^n}\big(e^{-t^{n-1}C_{n-1}}\cdots e^{-t^{2}C_{2}}e^{-tX}e^{-tY}e^{t(X+Y)}\big)\Big]_{t=0}\;.$$ I am aware of a couple of variations of this decomposition, e.g., in which the right-hand side of $(1)$ is transposed, or a symmetric decomposition. However, I was wondering if there exists a valid decomposition of the left hand side of $(1)$, of the form: $$e^{t(X+Y)} = e^{tX}\prod_{n=2}^{\infty}e^{t^nD_n}\,e^{tY} \;,$$ for some set of operators $D_n$ (presumably (sums of) nested commutators of X and Y)?
Such a decomposition would be really useful for a problem that I'm studying at the moment, however, I have been unable to find any resources that discuss this in detail, so I fear that it is not possible. If anyone could enlighten me on this, it would be much appreciated.