Lemma 4

Tag 0AY5, permanent. Cite it as [Bourbaki, Tag 0AY5].

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

If there exists an edge of order $\geq 6$ , then $l = 2$ .

Indeed, let $\{i, j\}$ be such an edge. If $l > 2$ , one of the edges $i, j$ (say $i$ ) would be linked to a third vertex $j'$ , since $X$ is connected. In view of the relation $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ we would have

\sum_{k \neq i} q_{ik}^2 \geq \frac{1}{4} + (\frac{\sqrt{3}}{2})^2 = 1

which is impossible (Lemma 1).

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

Cited by

Lemma 9, § 4
Theorem 1, § 4

Cites

Lemma 1