  This study examines the properties of heavy quarkonia $X$ by treating them as
bound states of $Q$ and $\bar{Q}$ at the LO level within the NRQCD framework,
where $Q$ represents either a charm or a bottom quark. The branching ratios for
the radiative leptonic decays $X\rightarrow \gamma l^{+} l^{-}$ are revisited
and the angular and energy/momentum distributions of the final state particles
are analyzed in the rest frame of $X$. Furthermore, we apply Lorentz
transformations from the rest frame of $X$ to the center-of-mass frame of $l^+
l^-$ to establish the connection between the widths ${\Gamma_{X \rightarrow
\gamma l^{+} l^{-}}}$ and ${\Gamma_{X \rightarrow l^{+} l^{-}}}$. When
comparing the connection with those documented in the literature (divided by
$2\pi$) for various $X$ states, such as $J/\Psi$, $\Psi(2S)$, $\Upsilon(1S)$,
and $\Upsilon(2S)$, relative differences typically around or below 10\% can be
found, which is comparable to the NLO corrections of $O(\alpha)$ and $O(v^4)$.
However, we observe a significant disparity in the ratio between
${\Gamma_{\Psi(2S) \to \gamma \tau^+ \tau^-}}$ and ${\Gamma_{\Psi(2S) \to
\tau^+ \tau^-}}$, with our prediction being four times larger than those in the
literature. The outcomes derived from this study held practical implications in
describing the QED radiative processes and contribute to the investigation of
QCD processes associated with the decays of heavy quarkonia and the searches
for new physics.
