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In mathematics, an orthogonal symmetric Lie algebra is a pair {\displaystyle } consisting of a real Lie algebra g {\displaystyle {\mathfrak {g}}} and an automorphism s {\displaystyle s} of g {\displaystyle {\mathfrak {g}}} of order 2 {\displaystyle 2} such that the eigenspace u {\displaystyle {\mathfrak {u}}} 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 u {\displaystyle {\mathfrak {u}}} intersects the center of g {\displaystyle {\mathfrak {g}}} 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, s {\displaystyle s} being the differential of a symmetry.
Let {\displaystyle } be effective orthogonal symmetric Lie algebra, and let p {\displaystyle {\mathfrak {p}}} denotes the -1 eigenspace of s {\displaystyle s}. We say that {\displaystyle } is of compact type if g {\displaystyle {\mathfrak {g}}} is compact and semisimple. If instead it is noncompact, semisimple, and if g = u + p {\displaystyle {\mathfrak {g}}={\mathfrak {u}}+{\mathfrak {p}}} is a Cartan decomposition, then {\displaystyle } is of noncompact type. If p {\displaystyle {\mathfrak {p}}} is an Abelian ideal of g {\displaystyle {\mathfrak {g}}} , then {\displaystyle } is said to be of Euclidean type.
Every effective, orthogonal symmetric Lie algebra decomposes into a direct sum of ideals g 0 {\displaystyle {\mathfrak {g}}_{0}} , g − {\displaystyle {\mathfrak {g}}_{-}} and g + {\displaystyle {\mathfrak {g}}_{+}} , each invariant under s {\displaystyle s} and orthogonal with respect to the Killing form of g {\displaystyle {\mathfrak {g}}} , and such that if s 0 {\displaystyle s_{0}} , s − {\displaystyle s_{-}} and s + {\displaystyle s_{+}} denote the restriction of s {\displaystyle s} to g 0 {\displaystyle {\mathfrak {g}}_{0}} , g − {\displaystyle {\mathfrak {g}}_{-}} and g + {\displaystyle {\mathfrak {g}}_{+}} , respectively, then {\displaystyle } , {\displaystyle } and {\displaystyle } are effective orthogonal symmetric Lie algebras of Euclidean type, compact type and noncompact type.