Lemma 2

Tag 0AU7, permanent. Cite it as [Bourbaki, Tag 0AU7].

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

Let R be a root system in V. Let $(x|y)$ be a symmetric bilinear form on V, non-degenerate and invariant under W(R). Identify V with $V^*$ by means of this form. If $\alpha \in R$ , then $\alpha$ is non-isotropic and

\alpha^{\sim} = \frac{2\alpha}{(\alpha|\alpha)}.

This follows from formula (4) of Chap. V, § 2, no. 3.

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

Cited by

§ 1. Root systems
Corollary, § 1
Proposition 1, § 1
Proposition 4, § 1
Lemma 4, § 2

Cites

Chap. V, § 2, no. 3