  In 2020, a di-$J/\psi$ resonance centered at around 6.9 $\mathrm{GeV}$ was
discovered by the LHCb Collaboration, and was later confirmed by the ATLAS and
CMS Collaborations. In addition to the 6.9-$\mathrm{GeV}$ resonance, its low
energy counterpart centered at around 6552 $\mathrm{MeV}$ was also observed by
the CMS Collaboration. In this paper, we calculate the inclusive decay width of
such exotic mesons into light hadrons and into open charms, respectively,
assuming them as either genuine tetraquarks or molecules composed of two
$J/\psi$'s. Since the spins of these resonances have not yet been
experimentally measured, regarding them as S-wave bound states, we consider all
possible spin configurations, namely spin-0 and spin-2, that decay exclusively
into double $J/\psi$, and find that the ratio ($\mathcal{R}$) of the decay
width for the light-hadron final states to that for the charmed final states
differs significantly among different spin configurations, which can help to
explore the nature of the observed di-$J/\psi$ resonances and fix the mixing
angle for the $J^{PC}=0^{++}$ tetraquark.
