  The recently observed $\alpha$-decay chain
${}^{108}\text{Xe}\to{}^{104}\text{Te}\to{}^{100}\text{Sn}$ [K.~Auranen
\emph{et al.}, Phys.\ Rev.\ Lett.\ {\bf121}, 182501 (2018)] could provide
valuable information on the $\alpha$ clustering in even-even nuclei slightly
above the major shells $Z=50$ and $N=50$. In this work, this $\alpha$-decay
chain is studied theoretically within the framework of the density-dependent
cluster model plus the two-potential approach. We calculate the $\alpha$-decay
half-lives of ${}^{104}$Te and ${}^{108}$Xe and study in detail their
dependence on the $Q$ value and the density-profile parameters of the core
nucleus. Various physical properties of ${}^{104}$Te, the heaviest nucleus with
a doubly magic self-conjugate core $+$ $\alpha$, are calculated, with two
different assumptions on the renormalization factor of the double-folding
potential, which could be a useful reference for future experimental studies.
