Let $\mathfrak g$ be a finite-dimensional real [Lie algebra](/page/Lie%20Algebra), and let $\operatorname{BCH}(X,Y)$ denote the Dynkin Baker--Campbell--Hausdorff Lie series wherever it converges. There exist an open neighbourhood $U \subset \mathfrak g$ of $0$, an open neighbourhood $\Omega \subset U \times U$ of $(0,0)$, and a smooth map $*: \Omega \to U$ such that, for every $(X,Y)\in \Omega$, the BCH series converges and $X*Y=\operatorname{BCH}(X,Y)$. The neighbourhoods may be chosen so that this operation is a local Lie group law: $(0,X),(X,0)\in\Omega$ imply $0*X=X$ and $X*0=X$; if $X,-X\in U$ and $(X,-X),(-X,X)\in\Omega$, then $X*(-X)=0=(-X)*X$; and for all $X,Y,Z\in U$ for which $(X,Y),(Y,Z),(X*Y,Z),(X,Y*Z)\in\Omega$, associativity holds: $(X*Y)*Z=X*(Y*Z)$.