  Let $E/F$ be a quadratic extension of a non-Archimedian local field.
Splitting of the 2-fold metaplectic cover of ${\rm Sp}_{2n}(F)$ when restricted
to various subgroups of ${\rm Sp}_{2n}(F)$ plays an important role in
application of the Weil representation of the metaplectic group. In this paper
we prove the splitting of the metaplectic cover of ${\rm GL}_{2}(E)$ over the
subgroups ${\rm GL}_{2}(F)$ and $D_{F}^{\times}$, where $D_{F}$ is the
quaternion division algebra with center $F$, as a first step in our study of
the restriction of representations of metaplectic cover of ${\rm GL}_{2}(E)$ to
${\rm GL}_{2}(F)$ and $D_{F}^{\times}$. These results were suggested to the
author by Professor Dipendra Prasad.
