Let $G$ denote a connected Lie group and $H$ a closed subgroup. Suppose that $\sigma$ is an involutive automorphism of $G$. Assume that $(G,H,\sigma)$ is a Riemannian symmetric pair.
So far I have only encountered examples of Riemannian symmetric pairs where the closed subgroup $H$ is already compact. Is there a (possibly) easy example of a Riemannian symmetric pair where the group $H$ is non-compact?
There are plenty of such examples. Just consider the quotient space $V/W$ of Euclidean vector space $V$ over its subspace $W$. The Riemannian symmetric pair you are looking for is $(V,W,-\text{id})$.