  The purpose of this paper is to study algebras of singular integral operators
on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the
composition of Calder\'on-Zygmund operators with different homogeneities, such
as operators that occur in sub-elliptic problems and those arising in elliptic
problems. For example, one would like to describe the algebras containing the
operators related to the Kohn-Laplacian for appropriate domains, or those
related to inverses of H\"ormander sub-Laplacians, when these are composed with
the more standard class of pseudo-differential operators. The algebras we study
can be characterized in a number of different but equivalent ways, and consist
of operators that are pseudo-local and bounded on $L^{p}$ for $1<p<\infty$.
While the usual class of Calder\'on-Zygmund operators is invariant under a
one-parameter family of dilations, the operators we study fall outside this
class, and reflect a multi-parameter structure.
