  We introduce the notion of a quasi-connected reductive group over a field of
characteristic 0 to be an almost direct product of a connected semisimple group
and a quasi-torus (a group of multiplicative type). We show that a linear
algebraic group is reductive and quasi-connected if and only if it is
isomorphic to a normal subgroup of a connected reductive group. We compute the
first Galois cohomology set H^1(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.
