A Splitting into the Double Cover of SL(3,$\mathbb{R}$)

We provide a formula for the splitting of a congruence subgroup of SL$(3,\mathbb{R})$ into the double cover of SL$(3,\mathbb{R})$ in terms of Pl\"{u}cker coordinates and prove that the splitting satisfies a twisted multiplicativity. The existence of this splitting and a formula (in terms of a different set of coordinates) was proved by S.D. Miller in an unpublished note; the formula in terms of Pl\"{u}cker coordinates is advantageous to the computation of the Fourier coefficients of an Eisenstein series on the double cover of SL$(3)$ over $\mathbb{Q}$. The computation of these Fourier coefficients will be addressed in a forthcoming work.