Real homogenous spaces, Galois cohomology, and Reeder puzzles

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X=G/H. We ask: How many connected components has X(R)? We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles.
Comment: 42 pages. The final version, to appear in J. Algebra