Bourbaki

Theorem 3

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

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

If R is an irreducible reduced root system, its Dynkin graph is isomorphic to one of the graphs represented by the following diagrams:

Al∘−⋯−∘(l≥1 vertices )Bl∘−⋯−∘−∘(l≥2 vertices )Cl∘−⋯−∘−∘(l≥3 vertices )Dl∘−⋯−∘−∘−∘(l≥4 vertices )E6∘−⋯−∘E7∘−⋯−∘−∘−∘E8∘−⋯−∘−∘−∘−∘F4∘−∘−∘−∘G2∘−∘−∘ \begin{array}{ll} A_l & \circ-\cdots-\circ \quad (l \geq 1 \text{ vertices }) \\ B_l & \circ-\cdots-\circ-\circ \quad (l \geq 2 \text{ vertices }) \\ C_l & \circ-\cdots-\circ-\circ \quad (l \geq 3 \text{ vertices }) \\ D_l & \circ-\cdots-\circ-\circ-\circ \quad (l \geq 4 \text{ vertices }) \\ E_6 & \circ-\cdots-\circ \\ E_7 & \circ-\cdots-\circ-\circ-\circ \\ E_8 & \circ-\cdots-\circ-\circ-\circ-\circ \\ F_4 & \circ-\circ-\circ-\circ \\ G_2 & \circ-\circ-\circ \end{array}

No two of these graphs are isomorphic and there are irreducible reduced root systems having each of them as their Dynkin graph (up to isomorphism).

The first assertion follows immediately from Th. 1, in view of the preceding remarks, the fact that the Coxeter groups of the graphs H_3, H_4 and I_2(p) (for p = 5 and p \geq 7) are not crystallographic, and the fact that the two possible inequalities for the double (resp. triple) bond of the Dynkin graph associated to the Coxeter graph F_4 (resp. G_2) give isomorphic Dynkin graphs. The second assertion is clear and the third will follow from the explicit construction of an irreducible reduced root system for each type, a construction that will be carried out in Nos. 5 to 13.

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

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Transcribed from the printed volume. Not translated.