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Based on $(106.41 \pm 0.86)\times 10^{6}$ $\psi(3686)$ events collected withthe BESIII detector at the BEPCII collider, the branching fractions of$\psi(3686) \to \pi^+\pi^- J/\psi$, $J/\psi \to e^+e^- $, and $J/\psi \to\mu^+\mu^-$ are measured. We obtain ${\cal B}[\psi(3686) \to\pi^+\pi^-J/\psi]=(34.98\pm 0.02\pm 0.45)%$, ${\cal B}[J/\psi \to e^+e^-] =(5.983 \pm 0.007 \pm 0.037)%$ and ${\cal B}[J/\psi \to \mu^+\mu^-] = (5.973 \pm0.007 \pm 0.038)%$. The measurement of ${\cal B}[\psi(3686) \to\pi^{+}\pi^{-}J/\psi]$ confirms the CLEO-c measurement, and is apparentlylarger than the others. The measured $J/\psi$ leptonic decay branchingfractions agree with previous experiments within one standard deviation. Theseresults lead to ${\cal B}[J/\psi \to l^+l^-] = (5.978 \pm 0.005 \pm 0.040)%$ byaveraging over the $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ channels and a ratio of${\cal B}[J/\psi \to e^+e^-] / {\cal B}[J/\psi \to \mu^+\mu^-] = 1.0017 \pm0.0017 \pm 0.0033$, which tests $e$-$\mu$ universality at the four tenths of apercent level. All the measurements presented in this paper are the mostprecise in the world to date.

Precision Measurements of ${\cal B}[\psi(3686) \to \pi^+\pi^- J/\psi]$ and ${\cal B}[J/\psi\to l^+l^-]$