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Let $M$ be a compact Riemannian symmetric space. By the classification of Cartan, it belongs to the table of homogeneous spaces given in the Wikipedia page: https://en.wikipedia.org/wiki/...

Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...