  We record a lifting theorem for the intertwiner of two
$S_{\Omega}$-isometries which are those subnormal operator tuples whose minimal
normal extensions have their Taylor spectra contained in the Shilov boundary of
a certain function algebra associated with $\Omega$, $\Omega$ being a bounded
convex domain in $\C^n$ containing the origin. The theorem captures several
known lifting results in the literature and yields interesting new examples of
liftings as a consequence of its being applicabile to Cartesian products
$\Omega$ of classical Cartan domains in $\C^n$. Further, we derive intrinsic
characterizations of $S_{\Omega}$-isometries where $\Omega$ is a classical
Cartan domain of any of the types I, II, III and IV, and we also provide a neat
description of an $S_{\Omega}$-isometry in case $\Omega$ is a finite Cartesian
product of such Cartan domains.
