TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cartan subalgebras are conjugate in a complex semisimple Lie algebra

Statement

Any two Cartan subalgebras of a complex semisimple Lie algebra are conjugate under the identity component of the automorphism group of $g$ .

Facts & Assumptions

Given: Two Cartan subalgebras $h_{1}, h_{2}$ of a complex semisimple Lie algebra $g$ .

[F1]

For the connected adjoint complex algebraic group $G_{a d}$ with Lie algebra $g$ , Cartan subalgebras of $g$ are exactly the Lie algebras of maximal tori of $G_{a d}$ .

[F2]

Any two maximal tori of a connected complex algebraic group are conjugate.

Proof

technique · direct

1.1

By [F1], choose maximal tori $T_{1}, T_{2} \leq G_{a d}$ with $Lie (T_{i}) = h_{i}$ .

F1givenchoose

2.1

By [F2], some $g \in G_{a d}$ satisfies $g T_{2} g^{- 1} = T_{1}$ . Taking Lie algebras gives $Ad (g) (h_{2}) = h_{1}$ .

F2step 1.1algebra

3.1

The adjoint group is the identity component of the inner automorphism group of $g$ , so $Ad (g)$ is an automorphism in the identity component. This proves the claimed conjugacy.

step 2.1∎

Depends on

Used by

Regular semisimple elements form a dense open subset Lemma

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)