In this paper, the dilepton electromagnetic decays $chi_{cJ}(1P) to J/psi e^+e^-$ and $chi_{cJ}(1P) to Jpsi mu^+mu^-$, 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 $mathcal{B}[chi_{c2}(1P) to J/psi mu^+mu^-]=1.66^{+0.19}_{-0.19} times 10^{-4}$.
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