  Based on $(106.41 \pm 0.86)\times 10^{6}$ $\psi(3686)$ events collected with
the 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 ${\cal B}[J/\psi \to \mu^+\mu^-] = (5.973 \pm
0.007 \pm 0.038)%$. The measurement of ${\cal B}[\psi(3686) \to
\pi^{+}\pi^{-}J/\psi]$ confirms the CLEO-c measurement, and is apparently
larger than the others. The measured $J/\psi$ leptonic decay branching
fractions agree with previous experiments within one standard deviation. These
results lead to ${\cal B}[J/\psi \to l^+l^-] = (5.978 \pm 0.005 \pm 0.040)%$ by
averaging over the $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ channels and a ratio of
${\cal B}[J/\psi \to e^+e^-] / {\cal B}[J/\psi \to \mu^+\mu^-] = 1.0017 \pm
0.0017 \pm 0.0033$, which tests $e$-$\mu$ universality at the four tenths of a
percent level. All the measurements presented in this paper are the most
precise in the world to date.
