Corollary

Tag 0AUG, permanent. Cite it as [Bourbaki, Tag 0AUG].

Lie Groups and Lie Algebras, chapter VI, § 1. Root systems

Let R be a root system in V. The following conditions are equivalent: (i) R is irreducible; (ii) the $W(R)$ *-module* V is simple; (iii) the $W(R)$ *-module* V is absolutely simple. (ii) $\Longleftrightarrow$ (i): this follows from Prop. 5 and Maschke's Theorem (Chap. V, Appendix, Prop. 2). (iii) $\Longleftrightarrow$ (ii): this follows from Prop. 1 of Chap. V, § 2, no. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 11, § 1
Proposition 12, § 1
Proposition 13, § 1
Proposition 2, § 2

Cites

Chap. V, Appendix, Prop. 2
Prop. 1 of Chap. V, § 2, no. 1
Prop. 5