Cartan Decomposition - Cartan Involutions On Lie Algebras

Cartan Involutions On Lie Algebras

Let be a real semisimple Lie algebra and let be its Killing form. An involution on is a Lie algebra automorphism of whose square is equal to the identity. Such an involution is called a Cartan involution on if is a positive definite bilinear form.

Two involutions and are considered equivalent if they differ only by an inner automorphism.

Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.