  In this paper, we determine a basis for the on-shell 4-point amplitudes
$VVVV$ for Standard Model gauge bosons $V=W^\pm, Z, \gamma, g$. Following
previous work, this completes the analysis of 3- and 4-point amplitudes for the
Standard Model and could be used for model independent searches for beyond the
Standard Model physics at colliders. Our results include a Lagrangian
parametrization for the "primary" amplitudes, where including additional
derivatives leads to the Mandelstam "descendant" amplitudes, and upper bounds
on the coupling constants from imposing unitarity. We also perform an estimate
for the sensitivity for new $Z$ decays at the HL-LHC, finding that $Z\to \gamma
\bar{\ell}\ell$ could be searched for, but that other decay modes, like $Z\to
(\gamma\gamma\gamma, \gamma gg)$, are too small to be discovered after imposing
unitarity constraints.
