Theorem 1
Tag 0AYC, permanent. Cite it as [Bourbaki, Tag 0AYC].
Lie Groups and Lie Algebras, chapter VI, § 4. Classification of root systems
The graph of any irreducible finite Coxeter system (W, S) is isomorphic to one of the following:
A_l (l \geq 1\ vertices) B_l (l \geq 2\ vertices) D_l (l \geq 4\ vertices) E_6 E_7 E_8 F_4 4 G_2 6 H_3 5 H_4 5 I_2(p) p (p = 5\ or\ p \geq 7).
No two of these graphs are isomorphic.
Indeed, let W = (m_{ij}) be the Coxeter matrix of (W, S), and let l = Card(S). If one of the m_{ij} is \geq 6, then l = 2 (Lemma 4) and the Coxeter graph of (W, S) is of type G_2 or I_2(p) with p \geq 7. Assume now that all the m_{ij} \leq 5.
a) If the m_{ij} are not all equal to 3, the graph X of (W, S) is a chain and exactly one of the m_{ij} is equal to 4 or 5 (Lemma 7). If one of the m_{ij} is equal to 5, Lemma 9 shows that we have one of the types H_3, H_4 or I_2(5). If one of the m_{ij} is equal to 4, Lemma 9 shows that we have one of the types B_l or F_4.
b) Assume now that all the m_{ij} are equal to 3. If X is a chain, the Coxeter graph is of type A_l. If not, Lemma 10 shows that it is of type E_6, E_7, E_8 or D_l.
The fact that no two of the Coxeter graphs listed are isomorphic is clear.
Conversely:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.