  The production of a neutral and a charged vector boson with subsequent decays
into three charged leptons and a neutrino is a very important process for
precision tests of the Standard Model of elementary particles and in searches
for anomalous triple-gauge-boson couplings. In this article, the first
computation of next-to-leading-order electroweak corrections to the production
of the four-lepton final states $\mu^+\mu^- e^+ \nu_e$, $\mu^+\mu^- e^-
\bar\nu_e$, $\mu^+\mu^- \mu^+\nu_\mu$, and $\mu^+\mu^- \mu^- \bar\nu_\mu$ at
the Large Hadron Collider is presented. We use the complete matrix elements at
leading and next-to-leading order, including all off-shell effects of
intermediate massive vector bosons and virtual photons. The relative
electroweak corrections to the fiducial cross sections from quark-induced
partonic processes vary between $-3\%$ and $-6\%$, depending significantly on
the event selection. At the level of differential distributions, we observe
large negative corrections of up to $-30\%$ in the high-energy tails of
distributions originating from electroweak Sudakov logarithms. Photon-induced
contributions at next-to-leading order raise the leading-order fiducial cross
section by $+2\%$. Interference effects in final states with equal-flavour
leptons are at the permille level for the fiducial cross section, but can lead
to sizeable effects in off-shell sensitive phase-space regions.
