  In this paper, the dilepton electromagnetic decays $\chi_{cJ}(1P) \to J/\psi
e^+e^-$ and $\chi_{cJ}(1P) \to J\psi \mu^+\mu^-$, where $\chi_{cJ}$ denotes
$\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$, are calculated systematically in the
improved Bethe-Salpeter method. The numerical results of decay widths and the
invariant mass distributions of the final lepton pairs are given. The
comparison is made with the recently measured experimental data of BESIII. It
is shown that for the cases including $e^+e^-$, the gauge invariance is
decisive and should be considered carefully. For the processes of
$\chi_{cJ}(1P) \to J/\psi e^+e^-$, the branching fraction are:
$\mathcal{B}[\chi_{c0}(1P) \to J/\psi e^+e^-]=1.06^{+0.16}_{-0.18} \times
10^{-4}$, $\mathcal{B}[\chi_{c1}(1P) \to J/\psi e^+e^-]=2.88^{+0.50}_{-0.53}
\times 10^{-3}$, and $\mathcal{B}[\chi_{c2}(1P) \to J/\psi
e^+e^-]=1.74^{+0.22}_{-0.21} \times 10^{-3}$. The calculated branching
fractions of $\chi_{cJ}(1P)\to J/\psi \mu^+\mu^-$ channels are:
$\mathcal{B}[\chi_{c0}(1P) \to J/\psi \mu^+\mu^-]=3.80^{+0.59}_{-0.64} \times
10^{-6}$, $\mathcal{B}[\chi_{c1}(1P) \to J/\psi
\mu^+\mu^-]=2.04^{+0.36}_{-0.38} \times 10^{-4}$, and
$\mathcal{B}[\chi_{c2}(1P) \to J/\psi \mu^+\mu^-]=1.66^{+0.19}_{-0.19} \times
10^{-4}$.
