Baker–Campbell–Hausdorff formula

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and exponential map $\exp:\mathfrak{g}\to G$. For $X,Y\in\mathfrak{g}$ sufficiently small, there is a unique $Z\in\mathfrak{g}$ near $0$ such that $\exp(X)\exp(Y)=\exp(Z)$; write $Z=\mathrm{BCH}(X,Y)$.

Theorem (BCH). In a neighborhood of $0\in\mathfrak{g}$,

where the omitted terms are (universal) Lie polynomials in iterated brackets of total degree $\ge 4$.

Moreover, if $\mathfrak{g}$ is nilpotent , then all sufficiently deep iterated brackets vanish and the BCH series truncates to a finite sum.

Context. BCH is the mechanism by which $\mathfrak{g}$ determines the local group law: via the logarithm map (local inverse to $\exp$), it turns multiplication in $G$ into an explicit Lie series on $\mathfrak{g}$. This is central to the Lie correspondence and to computations in exponential coordinates, especially for nilpotent and solvable groups.