Lemma 3

Tag 00XU, permanent. Cite it as [Bourbaki, Tag 00XU].

Lie Groups and Lie Algebras, chapter VIII, § 1. The Lie algebra sl(2, k) and its representations

If $x$ is an element of weight $\lambda$ , then $X_+x$ is an element of weight $\lambda + 2$ and $X_-x$ is an element of weight $\lambda -2$ .

Indeed, $HX_+x= [H, X_+]x+X_+Hx= 2X_+x+X_+\lambda x= (\lambda + 2)X_+x$ , and similarly $HX_-x= (\lambda -2)X_-x$ (cf. also Chap. VII, §1, no. 3, Prop. 10 (ii)).

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

Definition 1, § 1
Proposition 1, § 1

Cites

Chap. VII, §1, no. 3, Prop. 10