  We study the structure of C*-algebras associated with compactly aligned
product systems over group embeddable right LCM-semigroups. Towards this end we
employ controlled maps and a controlled elimination method that associates the
original cores to those of the controlling pair, and we combine with
applications of the C*-envelope theory for cosystems of nonselfadjoint operator
algebras recently produced. We derive several applications of these methods
that generalize results on single C*-correspondences.
  First we show that if the controlling group is exact then the co-universal
C*-algebra of the product system coincides with the quotient of the Fock
C*-algebra by the ideal of strong covariance relations. We show that if the
controlling group is amenable then the product system is amenable. In
particular if the controlling group is abelian then the co-universal C*-algebra
is the C*-envelope of the tensor algebra.
  Secondly we give necessary and sufficient conditions for the Fock C*-algebra
to be nuclear and exact. When the controlling group is amenable we completely
characterize nuclearity and exactness of any equivariant injective
Nica-covariant representation of the product system.
  Thirdly we consider controlled maps that enjoy a saturation property. In this
case we induce a compactly aligned product system over the controlling pair
that shares the same Fock representation, and preserves injectivity. By using
co-universality, we show that they share the same reduced covariance algebras.
If in addition the controlling pair is a total order then the fixed point
algebra of the controlling group induces a super-product system that has the
same reduced covariance algebra and is moreover reversible.
