Explicit formula for the $({\text{GL}}_2,{\text{GL}}_2)$ theta lift via Bruhat decomposition

Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $({\text{GL}}_2,{\text{GL}}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $({\text{GL}}_2(K),K^{\times })$ theta correspondence.