  We prove the existence of GSpin-valued Galois representations corresponding
to cohomological cuspidal automorphic representations of general symplectic
groups over totally real number fields under the local hypothesis that there is
a Steinberg component. This confirms the Buzzard-Gee conjecture on the global
Langlands correspondence in new cases. As an application we complete the
argument by Gross and Savin to construct a rank seven motive whose Galois group
is of type G_2 in the cohomology of Siegel modular varieties of genus three.
Under some additional local hypotheses we also show automorphic multiplicity
one as well as meromorphic continuation of the spin L-functions.
