Exercise 8, § 1

Let G be a compact semi-simple Lie group, $u$ an automorphism of G of order two, K the identity component of the set of fixed points of $u$, and X the manifold $G/K$ (the homogeneous space X is then said to be symmetric). a) Show that, if G is almost simple, then K is maximal among the connected closed subgroups of G distinct from G; in other words (Chap. III, §3, Exerc. 8), the operation of G on X is primitive. The symmetric space X is then said to be irreducible.

b) Assume that the manifold X is simply-connected; show that it is then isomorphic to a product of manifolds each of which is isomorphic either to a Lie group or to an irreducible symmetric space.