Let $A$ be an abelian variety over a field of characteristic $0$ and let $R \subset \mathrm{End}(A)$ be a (commutative, if that matters) subring. Suppose that $B \subset A$ is an $R$-stable abelian subvariety. Does $B$ have an $R$-stable complement? More precisely, does there exist an $R$-stable abelian subvariety $C \subset A$ such that $B + C = A$ and $B \cap C$ is finite?
I am guessing that a $C$ does not exist in general, but I do not have an example. In the positive direction, when $R = \mathbb{Z}[G]$ is a group ring, I think $C$ should exist by building a $G$-equivariant polarization and running the usual argument with the induced polarization on $B$ (Proposition 13.5.1 in the book "Complex abelian varieties" of Birkenhake and Lange is a result of this flavor).