  Let $X$ (resp. $Y$) be a curve of genus 1 (resp. 2) over a base field $k$
whose characteristic does not equal 2. We give criteria for the existence of a
curve $Z$ over $k$ whose Jacobian is up to twist (2,2,2)-isogenous to the
products of the Jacobians of $X$ and $Y$. Moreover, we give algorithms to
construct the curve $Z$ once equations for $X$ and $Y$ are given. The first of
these involves the use of hyperplane sections of the Kummer variety of $Y$
whose desingularization is isomorphic to $X$, whereas the second is based on
interpolation methods involving numerical results over $\mathbb{C}$ that are
proved to be correct over general fields a posteriori. As an application, we
find a twist of a Jacobian over $\mathbb{Q}$ that admits a rational 70-torsion
point.
