Remark 3

Tag 018O, permanent. Cite it as [Bourbaki, Tag 018O].

Lie Groups and Lie Algebras, chapter VIII, § 12. Chevalley orders

Let E be a finite dimensional $\mathfrak{g}$ -module, $\mathscr{E}$ an admissible lattice in E, $\mathfrak{c}$ the centre of $\mathfrak{g}$ . By Lemma 6, every element of $\mathfrak{c}$ defines a diagonalizable endomorphism of E. Hence E is semi-simple (Chap. I, §6, no. 5, Th. 4). Thus, E is a direct sum of simple $\mathscr{D}\mathfrak{g}$ -modules on which $\mathfrak{c}$ induces homotheties. By Lemma $6,\mathscr{E}=\oplus (\mathscr{E}\cap E^{\lambda})$ and, for all weights $\lambda$ of E, we have

\lambda (\mathscr{H})\subset \mathbf{Z}

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

Theorem 4, § 12

Cites

Chap. I, §6, no. 5, Th. 4 not in this corpus
Lemma 6