Galois cohomology and component group of a real reductive group

Let G be a connected reductive group over the field of real numbers ℝ. Using results of our previous joint paper, we compute combinatorially the first Galois cohomology set H 1 (ℝ, G ) in terms of reductive Kac labelings. Moreover, we compute the group of connected components π 0 G (ℝ) of the real Lie group G (ℝ) and the maps in exact sequences containing π 0 G (ℝ) and H 1 (ℝ, G ).