  To a pair of elliptic curves, one can naturally attach two K3 surfaces: the
Kummer surface of their product and a double cover of it, called the Inose
surface. They have prominently featured in many interesting constructions in
algebraic geometry and number theory. There are several more associated
elliptic K3 surfaces, obtained through base change of the Inose surface; these
have been previously studied by Kuwata. We give an explicit description of the
geometric Mordell-Weil groups of each of these elliptic surfaces in the generic
case (when the elliptic curves are non-isogenous). In the non-generic case, we
describe a method to calculate explicitly a finite index subgroup of the
Mordell-Weil group, which may be saturated to give the full group. Our methods
rely on several interesting group actions, the use of rational elliptic
surfaces, as well as connections to the geometry of low degree curves on cubic
and quartic surfaces. We apply our techniques to compute the full Mordell-Weil
group in several examples of arithmetic interest, arising from isogenous
elliptic curves with complex multiplication, for which these K3 surfaces are
singular.
