Galois cohomology of real quasi-connected reductive groups

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H¹(R, G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.