Classification of isotropic connected Riemannian manifolds

Question

This answer mentions a classification of all isotropic connected Riemannian manifolds up to isometry.

I am looking for a reference with a proof of this classification.

Answer 1

For a proof (which is long and technical) see Theorem 8.12.2 and Corollary 8.12.9 in

Wolf, Joseph A., Spaces of constant curvature. 3rd ed, Boston, Mass.: Publish or Perish, Inc. XV, 408 p. (1974). ZBL0281.53034.

Wolf also discussed some history of this classification in the end of section 8.12. There are two main parts to the argument: (a) every isotropic manifold is a symmetric space, (b) classifying symmetric isotropic spaces. Neither part is easy.