Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair consisting of a real Lie algebra and an automorphism of of order such that the eigenspace of s corresponding to 1 is a compact subalgebra. If "compactness" is omitted, it is called a symmetric Lie algebra. An orthogonal symmetric Lie algebra is said to be effective if intersects the center of trivially. In practice, effectiveness is often assumed; we do this in this article as well.
The canonical example is the Lie algebra of a symmetric space, being the differential of a symmetry.
Let be effective orthogonal symmetric Lie algebra, and let denotes the -1 eigenspace of. We say that is of compact type if is compact and semisimple. If instead it is noncompact, semisimple, and if is a Cartan decomposition, then is of noncompact type. If is an Abelian ideal of, then is said to be of Euclidean type.
Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals, and, each invariant under and orthogonal with respect to the Killing form of, and such that if, and denote the restriction of to, and, respectively, then, and are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.