Abelian varieties that split modulo all but finitely many primes

Let A be a simple abelian variety over a number field k such that End(A) is noncommutative. We show that A splits modulo all but finitely many primes of k. We prove this by considering the subalgebras of End(Aₚ) which have prime Schur index. Our main tools are Tate's characterization of endomorphism algebras of abelian varieties over finite fields, and a Theorem of Chia-Fu Yu on embeddings of simple algebras.