Finding the complement of an elliptic curve inside a Jacobian

This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $\pi :X\to E$ to an elliptic curve, determine an elliptic curve $E^{\mathrm{\prime }}$ and a map ${\pi }^{\mathrm{\prime }}:X\to E^{\mathrm{\prime }}$ independent of $\pi$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\mathrm{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here.