  Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable
integral representation on $\mathrm{GSp}_4$ 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
$\mathrm{PGL}_4$ representing the product of the $L$-functions attached to the
three fundamental representations of the Langlands $L$-group
$\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, is
a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGU}(2,2)$
representing the product of the degree 8 standard $L$-function and the degree 6
exterior square $L$-function.
