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 (i.e., the set  of fixed points) 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. References 

Lie algebras