CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Rank is well-defined

Statement

Assume the Axiom of Choice. All maximal tori of a compact connected Lie group have the same dimension; this common dimension is the rank of $G$ .

Facts & Assumptions

Given: Assume the Axiom of Choice and a compact connected Lie group $G$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it enters through the conjugacy theorem [L1].

[L1]

Maximal tori of $G$ exist, and any two maximal tori of $G$ are conjugate (Existence of maximal tori, Conjugacy of maximal tori).

[L2]

Conjugation $C_{g} (h) = g h g^{- 1}$ is a Lie-group automorphism, hence a diffeomorphism; its restriction to a maximal torus $T_{1}$ is a Lie-group isomorphism onto $g T_{1} g^{- 1}$ , so $\dim T_{1} = \dim g T_{1} g^{- 1}$ , and a Lie group and its Lie algebra have the same dimension (Conjugation and the adjoint representation of a Lie group, Tori and maximal tori).

Proof

technique · direct

1.1

By [L1] choose a maximal torus $T_{0} \leq G$ . If $T \leq G$ is any maximal torus, [L1] gives $g \in G$ with $T = g T_{0} g^{- 1}$ ; by [L2] conjugation restricts to a Lie-group isomorphism $T_{0} \to T$ , so $\dim T = \dim T_{0}$ .

L1L2

2.1

Maximal tori exist by step 1.1, and every maximal torus has the same dimension as $T_{0}$ . Defining the rank of $G$ to be this common dimension is therefore meaningful and unambiguous. The Axiom of Choice entered only through [L1].

A1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)