  We adopt the effective Lagrangian approach to study the strong decays of the
$1^-(0^{++})$ $D^\ast\bar{D}^\ast$ molecular state [denoted as
$T_{\psi0}^a(4010)$ according to the LHCb naming convention] through triangle
diagrams. The decay channels include the open-charm $D\bar{D}$, and the
hidden-charm $\eta_c\pi$, $J/\psi\rho$, and $\chi_{c1}\pi$. The coupling
between the $T_{\psi0}^a(4010)$ and its constituents $D^\ast\bar{D}^\ast$ is
obtained by solving for the residue of the scattering T-matrix at the pole. Our
calculation yields a total width of $(12.0$$-$$35.4)$ MeV for the
$T_{\psi0}^a(4010)$ state, with its main decay channels being $\eta_c\pi$ and
$\chi_{c1}\pi$. The $X(4100)$ and $X(4050)$ have similar masses and widths,
with both masses being close to the $D^\ast\bar{D}^\ast$ threshold.
Additionally, their decay final states are consistent with those of the
$T_{\psi0}^a(4010)$. Therefore, it is likely that they represent the same state
and both potentially correspond to the $T_{\psi0}^a(4010)$. We suggest that
future experiments focus on searching for the $T_{\psi0}^a(4010)$ signal in the
final states $\eta_c\pi^-$, $\chi_{c1}\pi^-$ and $D^0D^-$ of the
$B^0\to\eta_c\pi^-K^+$, $\chi_{c1}\pi^- K^+$ and $D^0D^-K^+$ processes,
respectively, as well as further investigating its resonance parameters with
Flatt\'e-like formula.
