Bourbaki

Remark 5

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

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

Let (Xα)α∈R(X_{\alpha})_{\alpha\in R} be the Chevalley system constructed above. If α,β,α+β∈R\alpha , \beta , \alpha +\beta \in R and if we put [Xα,Xβ]=Nα,βXα+β[X_{\alpha}, X_{\beta}] = N_{\alpha ,\beta}X_{\alpha+\beta}, we have [Xα,Xβ]∈[X_{\alpha}, X_{\beta}]\in G∩gα+β\mathscr{G}\cap \mathfrak{g}^{\alpha+\beta}, and we recover the fact that Nα,β∈ZN_{\alpha ,\beta}\in \mathbf{Z} (cf. §2, no. 4, Prop. 7).

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

Nothing.

Cites

Transcribed from the printed volume. Not translated.