Corollary 1

Tag 011P, permanent. Cite it as [Bourbaki, Tag 011P].

Lie Groups and Lie Algebras, chapter VIII, § 5. Automorphisms of a semi-simple Lie algebra

Let $(B,(X_{\alpha})_{\alpha\in B})$ be a framing of $(\mathfrak{g},\mathfrak{h})$ . Let G be the set of $s\in$ Aut( $\mathfrak{g},\mathfrak{h}$ ) that leave the framing invariant. Then Aut( $\mathfrak{g},\mathfrak{h}$ ) is the semi-direct product of G and $\varepsilon^{-1}(W(R))$ .

Indeed, $G\cap \varepsilon^{-1}(W(R)) =\{1\}$ by Prop. 1, and

Aut( $\mathfrak{g},\mathfrak{h}$ ) $= G.\varepsilon^{-1}(W(R))$

since $\varepsilon$ is surjective (Prop. 2).

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, § 5
Corollary 2, § 5

Cites

Prop. 1
Prop. 2