  We consider QCD radiative corrections to the production of four charged
leptons in hadron collisions. We present the computation of the next-to-leading
order QCD corrections to the loop-induced gluon fusion contribution. Our
predictions include, for the first time, also the quark-gluon partonic
channels. The computed corrections, which are formally of ${\cal O}(\alpha_{\rm
s}^3)$, turn out to increase the loop-induced Born-level result by an amount
ranging from 75% to 71% as $\sqrt{s}$ ranges from 8 to 13 TeV. We combine our
result with state-of-the-art NNLO corrections to the quark annihilation
channel, and present updated predictions for fiducial cross sections and
distributions for this process.
