Scholium

To any splittable semi-simple Lie algebra $\mathfrak{g}$ is associated a Dynkin graph, which determines $\mathfrak{g}$ up to isomorphism (Th. 2 (iii)). This graph is non-empty and connected if and only if $\mathfrak{g}$ is simple (§3, no. 2, Cor. 1 of Prop. 6). By Th. 1 of no. 3, and Chap. VI, §4, no. 2, Th. 3, the splittable simple Lie algebras are the algebras of type $A_l(l\geq 1), B_l(l\geq 2), C_l(l\geq 3), D_l(l\geq 4)$, $E_6, E_7, E_8, F_4, G_2$. No two algebras in this list are isomorphic.