It seems to me that Baker-Cambell-Hausdorff is just (noncommutative) algebra. A nice closed formula formula was found by Dynkin On a representation of the series $\log(e^x e^y)$ in non-commuting $x$ and $y$ via their commutators Math. Sb. 25 (67) 1949. The book "Methods of Noncommutative Analysis" by Nazaikinskii, Shatalov, and Sternin has a nice presentation.

There is also an interesting paper of Mosolova: New formula for $\log(e^B e^A)$ in terms of commutators of $A$ and $B$. Math. Notes. 29 1978.

It seems to me that Baker-Cambell-Hausdorff is just (noncommutative) algebra. A nice closed formula was found by Dynkin On a representation of the series $\log(e^x e^y)$ in non-commuting $x$ and $y$ via their commutators Math. Sb. 25 (67) 1949. The book "Methods of Noncommutative Analysis" by Nazaikinskii, Shatalov, and Sternin has a nice presentation.

There is also an interesting paper of Mosolova: New formula for $\log(e^B e^A)$ in terms of commutators of $A$ and $B$. Math. Notes. 29 1978.