Let $\mathbb F\in\{\mathbb R,\mathbb C\}$, let $\mathfrak g$ be a finite-dimensional [Lie algebra](/page/Lie%20Algebra) over $\mathbb F$, and let $G$ be a local Lie group integrating $\mathfrak g$ with identity element $e$. Let $\exp$ denote the local exponential map from a neighbourhood of $0\in\mathfrak g$ to a neighbourhood of $e\in G$. Choose neighbourhoods $U\subset \mathfrak g$ of $0$ and $V\subset G$ of $e$ such that $\exp:U\to V$ is a local diffeomorphism with local inverse $\log:V\to U$, and shrink $U$ so that $\exp(X)\exp(Y)\in V$ for all $X,Y\in U$ sufficiently close to $0$. Define the local Baker--Campbell--Hausdorff map $\operatorname{BCH}$ near $(0,0)\in\mathfrak g\times\mathfrak g$ by

\begin{align*} \operatorname{BCH}(X,Y)=\log(\exp(X)\exp(Y)). \end{align*}

For each $A\in\mathfrak g$, let $\operatorname{ad}_A:\mathfrak g\to\mathfrak g$ be the [linear map](/page/Linear%20Map) $W\mapsto [A,W]$. Then the local analytic expansion of $\operatorname{BCH}$ near $(0,0)$, truncated by homogeneous bracket degree through degree $3$, is

\begin{align*} \operatorname{BCH}(X,Y)=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[X,[X,Y]]-\frac{1}{12}[Y,[X,Y]]+O_4(X,Y). \end{align*}

Here $O_4(X,Y)$ denotes the part of the local analytic BCH expansion consisting of homogeneous Lie polynomials in $X$ and $Y$ of bracket degree at least $4$.