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Cartan subalgebras are conjugate in a complex semisimple Lie algebra
Statement
Assume the Axiom of Choice. Any two Cartan subalgebras of a finite-dimensional complex semisimple Lie algebra are conjugate under the identity component of the automorphism group of .
Facts & Assumptions
Given: The Axiom of Choice and two Cartan subalgebras of a finite-dimensional complex semisimple Lie algebra .
The Axiom of Choice is The Axiom of Choice.
Under [A1], two Cartan subalgebras are conjugate by an inner automorphism in the connected adjoint group (Conjugacy of Cartan subalgebras).
Proof
By [A1, F1], an element of the connected adjoint group carries to by its adjoint automorphism.
The continuous image of a connected group is connected and contains the identity. Therefore its adjoint automorphisms lie in the identity component of , proving the claim.
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Used by
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Sources
- Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)