Every semisimple Lie algebra g is classified by its Dynkin diagram. A Dynkin diagram is a graph with a few different kinds of possible edges. The connected components of the graph correspond to the irreducible subalgebras of g. So a simple Lie algebra's Dynkin diagram has only one component. The rules are restrictive. In fact, there are only certain possibilities for each component, corresponding to the classification of semi-simple Lie algebras.

Cartan matrix | Coxeter-Dynkin diagram | Killing form | Lie algebra | Lie algebra root | Lie group | root lattice | simple Lie algebra | Weyl group

Eugene Borisovich Dynkin