Theorem 4

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

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

Let (W, S) be an irreducible Coxeter system with S finite. The associated quadratic form (Chap. V, § 4, no. 1) is positive and degenerate if and only if the Coxeter graph of (W, S) is isomorphic to one of the following:

\begin{array}{ll} \tilde{A}_1 & \circ \longrightarrow \infty \\ \tilde{A}_l \quad (l \geq 2) & \circ \longrightarrow \cdots \longrightarrow \circ \quad \text{(circuit with } l + 1 \text{ vertices)} \\ \tilde{B}_2 & \circ \longrightarrow 4 \longrightarrow \circ \\ \tilde{B}_l \quad (l \geq 3) & \circ \longrightarrow \cdots \longrightarrow \circ \longrightarrow 4 \quad \text{(l + 1 vertices)} \\ \tilde{C}_l \quad (l \geq 3) & \circ \longrightarrow 4 \longrightarrow \cdots \longrightarrow 4 \quad \text{(l + 1 vertices)} \\ \tilde{D}_l \quad (l \geq 4) & \circ \longrightarrow \cdots \longrightarrow \circ \quad \text{(l + 1 vertices)} \\ \tilde{E}_6 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{E}_7 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{E}_8 & \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \longrightarrow \circ \\ \tilde{F}_4 & \circ \longrightarrow \circ \longrightarrow 4 \longrightarrow \circ \\ \tilde{G}_2 & \circ \longrightarrow 6 \longrightarrow \circ \end{array}

No two of these Coxeter graphs are isomorphic.

By Chap. V, § 4, no. 9 and Prop. 8 of § 2, no. 5, the Coxeter systems whose quadratic form is positive and degenerate are those which correspond to the affine Weyl groups of irreducible reduced root systems. The theorem therefore follows from the determination of the completed Dynkin graphs made in nos. 5 to 13 below.

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

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Nothing.

Theorem 4

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