  Building on work of Fayad and Nekov\'{a}\v{r}, we show that a certain part of
the etale cohomology of some abelian-type Shimura varieties is semisimple,
assuming the associated automorphic Galois representations exists, and
satisfies some good properties. The proof combines an abstract semisimplicity
criterion of Fayad-Nekov\'{a}\v{r} with the Eichler-Shimura relations.
