Galois representations for general symplectic groups

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 7 motive whose Galois group is of type G₂ in the cohomology of Siegel modular varieties of genus 3. Under some additional local hypotheses we also show automorphic multiplicity 1 as well as meromorphic continuation of the spin L-functions.