CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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The local Lie-group law is determined by the Lie bracket

Statement

Assume ${A C}_{ω}$ . In exponential coordinates near the identity of a finite-dimensional real Lie group, multiplication is

$(X, Y) ⟼ BCH (X, Y) .$

Thus the germ of multiplication at the identity is determined by the Lie-algebra bracket.

Facts & Assumptions

Given: ${A C}_{ω}$ , a finite-dimensional real Lie group, and the local logarithm and BCH neighborhood below.

[F1]

On a sufficiently small neighborhood, $\log_{G} (\exp_{G} X \exp_{G} Y) = BCH (X, Y)$ . Baker–Campbell–Hausdorff theorem.

[F2]

The local logarithm is inverse to the exponential on its stated domain. Local logarithm on a Lie group.

[F3]

Countable choice is the assumption inherited by both suppliers. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Proof

technique · direct

1.1

Choose the neighborhood supplied by [F1], already shrunk inside the domain in [F2]. In the chart $\log_{G}$ , the coordinate of the product of the points with coordinates $X$ and $Y$ is $\log_{G} (\exp_{G} X \exp_{G} Y) = BCH (X, Y)$ .

F1F2

2.1

Dynkin's series is built solely from addition, scalar multiplication, and the Lie bracket, so step 1.1 shows that the multiplication germ is determined by that bracket.

F1step 1.1

3.1

The identity makes the group nonempty. In dimensions zero and one the formula respectively reduces to the unique product and to $X + Y$ . Degenerate adjoint maps are allowed; no division by them occurs. There is no interval, endpoint, metric, or biconditional. ${A C}_{ω}$ is used exactly through [F1]–[F2], and the one neighborhood choice adds no family choice.

F1F2F3step 1.1step 2.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Michael Müger, Notes on the Baker-Campbell-Hausdorff-Dynkin theorem (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)