  Let $A$ be a simple abelian variety over a number field $k$ such that
$\operatorname{End}(A)$ is noncommutative. We show that $A$ splits modulo all
but finitely many primes of $k$. We prove this by considering the subalgebras
of $\operatorname{End}(A_{\mathfrak p})\otimes\mathbb{Q}$ 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.
