  We remove a parity condition from the construction of automorphic Galois
representations carried out in the Paris Book Project. We subsequently
generalize this construction to the case of `mixed-parity' (but still regular
essentially self-dual) automorphic representations over totally real fields,
finding associated geometric projective representations. Finally, we optimize
some of the Galois lifting results of arxiv:1207.6724 and apply them to the
previous mixed-parity setting.
