Orthogonal symmetric Lie algebra

An orthogonal symmetric Lie algebra is a pair \((\mathfrak{g}, s)\), where \(\mathfrak{g}\) is a real Lie algebra and \(s\) is an automorphism of order 2. The eigenspace \(\mathfrak{u}\) corresponding to eigenvalue 1 forms a compact subalgebra, with the term "symmetric Lie algebra" used if compactness isn't specified. Effectiveness requires that \(\mathfrak{u}\) intersects the center of \(\mathfrak{g}\) trivially, often assumed in practice.

The canonical example is the Lie algebra of a symmetric space, where \(s\) represents the differential of a symmetry. For an effective orthogonal symmetric Lie algebra, let \(\mathfrak{p}\) denote the -1 eigenspace of \(s\). The algebra is of compact type if \(\mathfrak{g}\) is compact and semisimple; noncompact type if \(\mathfrak{g}\) is noncompact, semisimple, and forms a Cartan decomposition with \(\mathfrak{u}\) and \(\mathfrak{p}\). If \(\mathfrak{p}\) is an Abelian ideal, it's of Euclidean type.

Every effective orthogonal symmetric Lie algebra decomposes into ideals \(\mathfrak{g}_0\), \(\mathfrak{g}_{-}\), and \(\mathfrak{g}_{+}\), each forming effective orthogonal symmetric Lie algebras of Euclidean, compact, and noncompact types respectively.