Galois cohomology of real semisimple groups

Let G be a connected, compact, semisimple algebraic group over the field of real numbers R. Using Kac diagrams, we describe combinatorially the first Galois cohomology sets H^1(R,H) for all inner forms H of G. As examples, we compute explicitly H^1 for all real forms of the simply connected group of type E_7 (which has been known since 2013) and for all real forms of half-spin groups of type D_{2k} (which seems to be new).