follows by an easy diagram chase.

This proposition leaves us some hope, because the $\beta$ in the proposition will not be an element of ${\operatorname{End}}B$ if $G\neq G'$. Also, if we consider the case $n=2$ and if the principally-polarized surface $(A,\lambda)$ is given to us explicitly as either a Jacobian or a product of polarized elliptic curves, then the theory of the Richelot isogeny will allow us to write down $(B,\mu)$ and $(B,\mu')$ explicitly as Jacobians, as we explain below. Thus, we would like to explicitly write down abelian surfaces $A$ with non-trivial automorphisms. In later sections we will consider two families of such explicitly-given surfaces: products of isogenous elliptic curves, and Jacobians with real multiplication by $\sqrt{2}$.

We close this section with a comment about Richelot duals and maximal isotropic subgroups. Suppose $C$ is a genus-$2$ curve defined by an equation $\delta y^2 = f$ and suppose $D$ is the Richelot dual of $C$ corresponding to a factorization $f = g_1 g_2 g_3$. Then there is an isogeny from the Jacobian of $C$ to the Jacobian of $D$ whose kernel is the order-$4$ subgroup $G$ of ${\operatorname{Jac}}C$ containing the classes of the divisors $(a_i,0) - (b_i,0)$, where $a_i$ and 