Corollary

Tag 0102, permanent. Cite it as [Bourbaki, Tag 0102].

Lie Groups and Lie Algebras, chapter VIII, § 3. Subalgebras of split semi-simple Lie algebras

Let $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$ .

(i) Every Cartan subalgebra of $\mathfrak{b}$ is a splitting Cartan subalgebra of $\mathfrak{g}$ .

(ii) If $\mathfrak{h}_1,\mathfrak{h}_2$ are Cartan subalgebras of $\mathfrak{b}$ , there exists $x\in [\mathfrak{b},\mathfrak{b}]$ such that $e^{ad_{\mathfrak{g}}x}\mathfrak{h}_1=\mathfrak{h}_2$ .

Assertion (i) follows from Prop. 9 (i) and Chap. VII, §3, no. 3, Prop. 3. Assertion (ii) follows from Prop. 9 (ii) and Chap. VII, §3, no. 4, Th. 3.

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

Corollary, § 3
Proposition 10, § 3

Cites

Chap. VII, §3, no. 3, Prop. 3
Chap. VII, §3, no. 4, Th. 3
Prop. 9