Classification of simple Lie algebras

Remarks

The proof has three steps. First, a simple Lie algebra is determined up to isomorphism by its root system, which is irreducible, by the theorem on the root system of a semisimple Lie algebra. Second, the connected Dynkin diagrams of root systems are exactly $A_l, B_l, C_l, D_l, E_6, E_7, E_8, F_4, G_2$; this is a combinatorial argument on positive definite quadratic forms. Third, each of these root systems comes from a simple Lie algebra. For the classical types one checks the matrix algebras special linear, orthogonal and symplectic, and in general Serre's theorem builds the algebra from its Cartan matrix. The bounds on $l$ remove repetitions: $B_1 \cong A_1$, $C_2 \cong B_2$, $D_3 \cong A_3$, while $D_1$ is abelian and $D_2 \cong A_1 \times A_1$ is not simple. The exceptional algebras are E8, E7, E6, F4 and G2. Over $\R$ or in positive characteristic the list is different.

Going deeper

For a student

Through the Lie correspondence, the theorem also classifies the simply connected complex simple Lie groups and the simply connected compact simple Lie groups. The simply laced diagrams of types $A$, $D$ and $E$ also classify the simple surface singularities and the quivers of finite representation type. In a course it is the endpoint of the structure theory: Cartan subalgebra, roots, root system, base, Cartan matrix, Dynkin diagram.