Baker–Campbell–Hausdorff formula

Statement

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 $0\in\mathfrak{g}$ ,

\mathrm{BCH}(X,Y) = X+Y+\frac12[X,Y]+\frac1{12}[X,[X,Y]]-\frac1{12}[Y,[X,Y]]+\cdots,

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.

Formal Lie series

The same expression exists without an analytic Lie group. Over $\mathbb Q$ , let $\widehat L(X,Y)$ be the completion by bracket degree of the free Lie algebra on $X,Y$ . In the completed universal enveloping algebra,

\operatorname{BCH}(X,Y) = \log\!\bigl(\exp(X)\exp(Y)\bigr) \in\widehat L(X,Y).

This identity defines a universal formal Lie series. Its homogeneous component of any fixed degree is a finite rational linear combination of iterated brackets, so it can be evaluated degree by degree in formal coordinates on every finite-dimensional Lie algebra over a characteristic-zero field.

More generally, the series evaluates in a complete filtered Lie algebra when brackets raise filtration and $X,Y$ lie in the positive filtration. In that setting “convergence” means convergence in the filtration, not convergence of real or complex numbers.

Associativity of multiplication of exponentials implies the formal identity

\operatorname{BCH}(\operatorname{BCH}(X,Y),Z) = \operatorname{BCH}(X,\operatorname{BCH}(Y,Z)).

Together with identity $0$ and inverse $-X$ , this makes BCH a formal group law and supplies the integration functor in the characteristic-zero formal Lie correspondence.

Analytic interpretation

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.

The formal identity and the analytic assertion have different hypotheses. Formal evaluation only uses degree completion and rational coefficients. Analytic evaluation requires a topology and sufficiently small inputs (unless nilpotence makes the series finite).

References

Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapters 1–3, Springer, 1989. Publisher record. Relevant: Chapter 2, exponential, logarithmic, and Hausdorff series.
Jean-Pierre Serre, Lie Algebras and Lie Groups, second edition, Springer, 1992. Publisher record. Relevant: Part I, formal Lie theory.