Bourbaki

Proposition 32

Tag 0AW9, permanent. Cite it as [Bourbaki, Tag 0AW9].

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

Assume that R is irreducible, and that all roots have the same length. Let α∈R\alpha \in R. The number of elements of R not orthogonal to α\alpha is 4h−64h - 6.

Let R′R' be the set of roots not proportional to and not orthogonal to α\alpha. By Chap. V, § 6, no. 2, Cor. of Th. 1,

(α∣α)2+(α∣−α)2+∑β∈R′(α∣β)2=h(α∣α)2, (\alpha | \alpha)^2 + (\alpha | -\alpha)^2 + \sum_{\beta \in R'} (\alpha | \beta)^2 = h(\alpha | \alpha)^2,

that is

∑β∈R′(α∣β)2=(h−2)(α∣α)2. \sum_{\beta \in R'} (\alpha | \beta)^2 = (h - 2)(\alpha | \alpha)^2.

If β∈R′\beta \in R', then (α∣β)=±12(α∣α)(\alpha | \beta) = \pm \frac{1}{2} (\alpha | \alpha) by the list in no. 3. Hence

14Card⁡R′=h−2,Card⁡R′=4h−8, \frac{1}{4} \operatorname{Card} R' = h - 2, \quad \operatorname{Card} R' = 4h - 8,

and the number of roots not orthogonal to α\alpha is Card⁡R′+2=4h−6\operatorname{Card} R' + 2 = 4h - 6.

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

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Cites

Transcribed from the printed volume. Not translated.