Cartan Decomposition - Cartan Pairs

Cartan Pairs

Let be an involution on a Lie algebra . Since, the linear map has the two eigenvalues . Let and be the corresponding eigenspaces, then . Since is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that

, and .

Thus is a Lie subalgebra, while any subalgebra of is commutative.

Conversely, a decomposition with these extra properties determines an involution on that is on and on .

Such a pair is also called a Cartan pair of .

The decomposition associated to a Cartan involution is called a Cartan decomposition of . The special feature of a Cartan decomposition is that the Killing form is negative definite on and positive definite on . Furthermore, and are orthogonal complements of each other with respect to the Killing form on .