Galois cohomology of reductive algebraic groups over the field of real numbers

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$of a connected reductive algebraic group $G$ over the field $\mathbb R$ of realnumbers in terms of a certain action of the Weyl group on the real points oforder dividing 2 of the maximal torus containing a maximal compact torus. Thisresult was announced with a sketch of proof in the author's 1988 note. Here wegive a detailed proof.Comment: V.1, v.2, v.3: 6 pages. V.4, v.5: 11 pages, the final version to appear in Communicationa in Mathematics. In this final version, Theorem 9 (the main result) of versions 1-3 became Theorem 3.