$b_i$ are the roots of $g_i$ in ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ (see [@Cassels-Flynn Ch. 9]). The subgroup $G$ is a maximal isotropic subgroup of the $2$-torsion of ${\operatorname{Jac}}C$ under the Weil pairing. Conversely, every $K$-defined maximal isotropic subgroup $G$ of $({\operatorname{Jac}}C)[2]$ arises in this way. Thus, given a $K$-defined maximal isotropic subgroup $G$, we can define the [*$G$-Richelot dual*]{} of $C$ to be the Richelot dual of $C$ with respect to the factorization $f = g_1 g_2 g_3$ that gives rise to $G$.

Proof of Theorem \[T-nonsimple2\] {#S-nonsimple}
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In this section we will prove Theorem \[T-nonsimple2\] by following the outline given in Section \[S-curves\] in the case where $A$ is a split abelian surface.

Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic not $2$ and suppose there is a $2$-isogeny $\psi$ from $E$ to $E'$. Let $Q$ be the nonzero element of $E[2](K)$ in the kernel of $\psi$, and let $P$ and $R$ be the other two geometric $2$-torsion points of $E$. Let $Q' = \psi(P) = \psi(R)$, so that $Q'$ is a nonzero element of $E'[2](K)$, and let $P'$ and $R'$ be the other geometric $2$-torsion points of $E'$. Suppose the discriminants of $E$ and $E'$ 