  We define a generic algorithmic framework to prove pure discrete spectrum for
the substitutive symbolic dynamical systems associated with some infinite
families of Pisot substitutions. We focus on the families obtained as finite
products of the three-letter substitutions associated with the multidimensional
continued fraction algorithms of Brun and Jacobi-Perron.
  Our tools consist in a reformulation of some combinatorial criteria
(coincidence conditions), in terms of properties of discrete plane generation
using multidimensional (dual) substitutions. We also deduce some topological
and dynamical properties of the Rauzy fractals, of the underlying symbolic
dynamical systems, as well as some number-theoretical properties of the
associated Pisot numbers.
