  We study genus 2 function fields with elliptic subfields of degree 2. The
locus $\L_2$ of these fields is a 2-dimensional subvariety of the moduli space
$\mathcal M_2$ of genus 2 fields. An equation for $\L_2$ is already in the work
of Clebsch and Bolza. We use a birational parameterization of $\L_2$ by affine
2-space to study the relation between the j-invariants of the degree 2 elliptic
subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find
a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic
subfields of degree 2; this family is parameterized by the j-invariant of these
subfields.
