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".