  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 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.
