How do I prove the statement that for a Riemannian symmetric space $M$, the isometry group $Iso(M)$ is a Lie group? What can we say about the dimension of $Iso(M)$? And is its action transitive on $M$? And what can we say about the isometry group of a Lie group endowed with a left/right (not bi) invariant Riemannian metric (as we know if the metric is bi invariant, then the Lie group is a symmetric space)?
P.S I not only look for answers but also for resource and literature to probe such related questions. (Note: In the definition of Riemannian symmetric space that I come across, the space is not assumed to be simply connected)