# Definition The Baker-Campbell-Hausdorff (BCH) is given by $ e^{X} e^{Y}=e^{X+Y+\frac{1}{2}[X, Y]+\frac{1}{12}[X,[X, Y]]-\frac{1}{12}[Y,[X, Y]]+\ldots} $ where $X, Y$ are [[Lie Algebra|Lie algebraic]] elements. It can be shown that the series is composed entirely of iterated [[Commutator|commutators]]. Since Lie algebras are closed under commutators, this means that the exponent on the right hand side is an element of the Lie algebra. For $X$ and $Y$ sufficiently small, the series converges. This formula is analogous to the [[Lie Product Formula]]. The Lie product formula expresses Lie algebra addition in terms of [[Lie group]] multiplication while the BCH formula expresses group multiplication in terms of the commutator of the Lie algebra. The BCH formula tells us that much of the group structure of $G$ is encoded in the commutator on $\mathfrak{g}$. # Proof