Multivariate Rankin-Selberg integrals on ${\mathrm{G}\mathrm{L}}_4$ and $\mathrm{G}\mathrm{U}(2,2)$

Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on ${\mathrm{P}\mathrm{G}\mathrm{L}}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group ${\mathrm{S}\mathrm{L}}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{P}\mathrm{G}\mathrm{U}(2,2)$ representing the product of the degree 8 standard $L$-function and the degree 6 exterior square $L$-function.