  We consider a family of Levi-degenerate finite type hypersurfaces in $\mathbb
C^2$, where in general there is no group structure. We lift these domains to
stratified Lie groups via a constructive proof, which optimizes the well-known
lifting procedure to free Lie groups of general manifolds defined by Rothschild
and Stein. This yields an explicit version of the Taylor expansion with respect
to the horizontal vector fields induced by the sub-Riemannian structure on
these hypersurfaces. Hence, as an application, we establish the Schatten class
estimates for the commutator of the Cauchy--Szeg\"{o} projection with respect
to a suitable quasi-metric defined on the hypersurface.
