Corollary 2

Tag 01J3, permanent. Cite it as [Bourbaki, Tag 01J3].

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

Aut( $\mathfrak{g}$ ) $=$ Aut $_0(\mathfrak{g})$ when $\mathfrak{g}$ is a splittable simple Lie algebra of type $A_1,B_n(n\geq 2)$ , $C_n(n\geq 2)$ , $E_7,E_8,F_4,G_2$ . The quotient Aut( $\mathfrak{g}$ ) $/$ Aut $_0(\mathfrak{g})$ is of order 2 when $\mathfrak{g}$ is of type $A_n(n\geq 2)$ , $D_n(n\geq 5)$ , $E_6$ ; it is isomorphic to $\mathfrak{S}_3$ when $\mathfrak{g}$ is of type $D_4$ .

This follows from Cor. 1 and Chap. VI, Plates I to IX.

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

Proposition 1, § 11
§ 13. Classical splittable simple Lie algebras

Cites

Cor. 1