this gives a direct sum decomposition with The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer is a Lie subalgebra of . The second condition means that is an -invariant complement to in . Thus any symmetric space is a reductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that brackets into . Conversely, given any Lie algebra with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on and minus the identity on , is an involutive automorphism. Riemannian symmetric spaces satisfy the Lie-theoretic characterization If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie group acting transitively on M (that is, M is Riemannian homogeneous). Therefore, if we fix some point p of M, M is diffeomorphic to the quotient G/K, where K denotes the isotropy group of the action of G on M at p. By differentiating the action at p we obtain an isometric action of K on TpM. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its 1-jet at any point) and so K is a subgroup of the orthogonal group of TpM, hence compact. Moreover, if we denote by sp: M → M the geodesic symmetry of M at p, the map is an involutive Lie group automorphism such that the isotropy group K is contained between the fixed point group and its identity component (hence an open subgroup) see the definition and following proposition on page 209, chapter IV, section 3 in Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information. To summarize, M is a symmetric space G/K with a compact isotropy group K. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a K-invariant inner product on the tangent space to G/K at the identity coset eK: such an inner product always exists by averaging, since K is compact, and by acting with G, we obtain a G-invariant Riemannian metric g on G/K. To show that G/K is Riemannian symmetric, consider any point p = hK (a coset of