A question on splitting of metaplectic covers

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.

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