Suppose I have $n$ operators $X_1,\ldots,X_n$, which are closed under commutators: $$[X_i,X_j]=c_1X_1+\cdots+c_nX_n,$$ where $c_j$ are constants (which will depend on $i$ and $j$).
Now consider a product of exponentials: $$\xi=\exp(\alpha_1X_1)\cdots\exp(\alpha_nX_n).$$ "Most of the time" I should also be able to write this a product of the same operators in a different order: $$\xi=\exp(\beta_1X_{\sigma(1)})\cdots\exp(\beta_nX_{\sigma(n)}),$$ where $\sigma$ is some permutation. My questions are:
- If it is possible to write $\xi$ as two different arrangements, is there a way to find the $\beta_i$ given $\sigma$, $\alpha_i$, and the algebra commutation relations?
- When can this fail? Is it possible that $\xi$ could be expressed as one ordering, but not another?
This question comes from considering the Wei-Norman expansion for matrix differential equations. These will have singularities at some points, but I wonder how much flexibility I have to jump between coordinate systems as I approach the singularities.