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.
Every orthogonal symmetric Lie algebra decomposes into a direct sum of ideals "of compact type", "of noncompact type" and "of Euclidean type".
