Riemannian symmetric pair of noncompact type

A Riemannian symmetric pair of noncompact type is a pair $(G,K)$ together with a global Cartan involution $\mathrm{\Theta }$ of $G$, in the following sense.

$G$ is a connected real semisimple Lie group with finite center and Lie algebra ${\mathfrak{g}}_0$, $\mathrm{\Theta }$ is a global Cartan involution of $G$ (Cartan involution of a real semisimple Lie algebra, Global Cartan decomposition for a connected finite center semisimple Lie group), $K=G^{\mathrm{\Theta }}$, and ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ is the Cartan decomposition attached to ${\theta }_{\ast }=d{\mathrm{\Theta }}_e$ (Cartan decomposition of a real semisimple Lie algebra). By Global Cartan decomposition for a connected finite center semisimple Lie group the subgroup $K$ is closed with Lie algebra ${\mathfrak{k}}_0$ and compact, so $K$ is a closed subgroup of $G$ and the coset space $G\mathrm{/}K$ is a homogeneous $G$-space (Homogeneous spaces of Lie groups); the pair is of noncompact type when, in addition, ${\mathfrak{g}}_0$ has no nonzero compact ideal (equivalently, when ${\mathfrak{p}}_0$ is not contained in any proper ideal of ${\mathfrak{g}}_0$, so that no compact factor of $G$ is carried along); this is the normalisation used below whenever a statement speaks of a symmetric pair of noncompact type, and without it the same construction applies after splitting off the compact ideals.

On the Lie-algebra level the same object is the pair $({\mathfrak{g}}_0,{\theta }_{\ast })$ consisting of a real semisimple Lie algebra ${\mathfrak{g}}_0$ and a Cartan involution ${\theta }_{\ast }$, with ${\mathfrak{g}}_0$ having no nonzero compact ideal; the corresponding symmetric space is $G\mathrm{/}K$ with the $G$-invariant Riemannian metric whose value at the origin $eK$ is the positive definite form $B_{{\theta }_{\ast }}$ restricted to ${\mathfrak{p}}_0\cong T_{eK}(G\mathrm{/}K)$. The metric, its invariance and its curvature are constructed in Cartan decomposition gives the invariant metric and curvature of G mod K, and the identification of ${\mathfrak{p}}_0$ with $G\mathrm{/}K$ is Cartan decomposition identifies p with the noncompact symmetric space.

The hypothesis that ${\mathfrak{g}}_0$ has no compact ideal is a normalisation used in the literature: ${\mathfrak{g}}_0$ is the direct sum of its compact and noncompact simple ideals, the compact ideals contribute a compact factor to $G$ that acts trivially on the symmetric space, and one removes them so that ${\mathfrak{p}}_0$ determines ${\mathfrak{g}}_0$ up to a compact summand.