Cartan subalgebra

To understand a complicated linear operator you diagonalize it; to understand a whole semisimple Lie algebra you find a maximal collection of elements you can diagonalize simultaneously. A Cartan subalgebra is exactly such a maximal mutually-diagonalizable abelian piece — the coordinate axes of the algebra. Decomposing the rest of the algebra into eigenspaces for this subalgebra is what produces the root system and unlocks the entire classification.

Abstractly, a Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H that equals its own normalizer: N_L(H) = H. For a semisimple Lie algebra over an algebraically closed field of characteristic zero, this simplifies to the concrete picture: H is a maximal abelian subalgebra all of whose elements act diagonalizably (are semisimple) under the adjoint representation. Such H always exist and are all conjugate, so their common dimension — the rank of L — is an invariant.

Given H, the adjoint action splits L into simultaneous eigenspaces: L = H (+) (sum over alpha of L_alpha), where each nonzero linear functional alpha on H appearing here is a root and L_alpha is its root space. This Cartan decomposition is the engine of structure theory: the roots form a root system, and almost everything about L is encoded in that finite set of vectors.