Theorem 1

Tag 00V8, permanent. Cite it as [Bourbaki, Tag 00V8].

Lie Groups and Lie Algebras, chapter VII, § 3. Conjugacy theorems

Assume that $k$ is algebraically closed. Let $\mathfrak{g}$ be a Lie algebra. The group E is normal in Aut( $\mathfrak{g}$ ) and operates transitively on the set of Cartan subalgebras of $\mathfrak{g}$ .

Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$ , and $v\in$ Aut( $\mathfrak{g}$ ). Then

vE(\mathfrak{h})v^{-1}= E(v(\mathfrak{h})) = E(\mathfrak{h})

so $E(\mathfrak{h}) = E$ is normal in Aut( $\mathfrak{g}$ ). If $\mathfrak{h}'$ is another Cartan subalgebra of $\mathfrak{g}$ , then, in the notation of Prop. $2,u^{'-1}u(\mathfrak{h}) =\mathfrak{h}'$ , and $u^{'-1}u\in E$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Theorem 1, § 3
Definition 1, § 2
Proposition 3, § 3
Theorem 2, § 3
Proposition 6, § 5
Theorem 2, § 5
Theorem 3, § 10
Proposition 3, § 2
Proposition 5, § 5
Theorem 1, § 8

Cites

Nothing.