Lemma 3

Tag 0AY4, permanent. Cite it as [Bourbaki, Tag 0AY4].

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

If $i$ belongs to 3 edges, these edges are of order 3.

If not, we would have, in view of the relation $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ ,

\sum_{j \neq i} q_{ij}^2 \geq \frac{1}{4} + \frac{1}{4} + (\frac{\sqrt{2}}{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.

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