Quadratic forms of signature $(2,2)$ or $(3,1)$ I: effective equidistribution in quotients of ${\mathrm{S}\mathrm{L}}_4(\mathbb{R})$

We prove an effective equidistribution theorem for orbits of horospherical subgroups of $\mathrm{S}\mathrm{O}(2,2)$ and $\mathrm{S}\mathrm{O}(3,1)$ in quotients of ${\mathrm{S}\mathrm{L}}_4(\mathbb{R})$ with a polynomial error term. In a forthcoming paper, we will use this theorem to prove an effective version of the Oppenheim conjecture for indefinite quadratic forms of signature $(2,2)$ or $(3,1)$ with a polynomial error rate.