  Using a data sample of $4.481\times 10^8 \psi^\prime$ events collected with
the BESIII detector, we report the first observation of the four-lepton-decays
$J/\psi\to e^+e^-e^+e^-$ and $J/\psi\to e^+e^-\mu^+\mu^-$ utilizing the process
$\psi^\prime\to \pi^+\pi^- J/\psi$. The branching fractions are determined to
be $[5.48\pm0.31~(\rm stat)\pm0.45~(\rm syst)]\times 10^{-5}$ and
$[3.53~\pm0.22~(\rm stat)\pm0.13~(\rm syst)]\times 10^{-5}$, respectively. The
results are consistent with theoretical predictions. No significant signal is
observed for $J/\psi\to \mu^+\mu^-\mu^+\mu^-$, and an upper limit on the
branching fraction is set at $1.6\times 10^{-6}$ at the 90$\%$ confidence
level. A CP asymmetry observable is constructed for the first two channels,
which is measured to be $(-0.012\pm0.054\pm0.010)$ and
$(0.062\pm0.059\pm0.006)$, respectively. No evidence for CP violation is
observed in this process.
