Corollary 1

Tag 013D, permanent. Cite it as [Bourbaki, Tag 013D].

Lie Groups and Lie Algebras, chapter VIII, § 7. Finite dimensional modules over a split semi-simple Lie algebra

The isomorphism $s^*$ takes the weights of V with respect to $\mathfrak{h}$ to those of V with respect to $\mathfrak{h}'$ ; corresponding weights have the same multiplicity.

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

Remark 2, § 7

Cites

Nothing.