Lemma 6

Tag 0AY7, permanent. Cite it as [Bourbaki, Tag 0AY7].

Lie Groups and Lie Algebras, chapter VI, § 4. Classification of root systems

If $\{i, j\}$ is of order 3, $W(M')$ is a finite Coxeter group.

Indeed, $q_{ij} = -\frac{1}{2}$ , so (1) becomes

\sum_{k,k' \in I'} q'_{kk'} \xi_k \xi_{k'} = \sum_{k,k' \in I} q_{kk'} \xi_k \xi_{k'}

and $(\xi_k)_{k \in I'} \mapsto \sum_{k,k' \in I'} q'_{kk'} \xi_k \xi_{k'}$ is a positive non-degenerate quadratic form. It now suffices to apply Th. 2 of Chap. V. § 4, no. 8.

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

Cited by

Lemma 7, § 4

Cites

§ 4, no. 8
Th. 2