  We apply a symbolic approach of the general quadratic decomposition of
polynomial sequences - presented in a previous article referenced herein - to
polynomial sequences fulfilling specific orthogonal conditions towards two
given functionals $u_{0},u_{1}$ belonging to the dual of the vector space of
polynomials with coefficients in $\mathbb{C}$. The general quadratic
decomposition produces four new sets of polynomials whose properties are
investigated with the help of the mentioned symbolic approach together with
further commands which inquire relevant features, as for instance, the
classical character of a polynomial sequence. The computational results are
detailed for a wide range of choices of parameters and co-recursive type
polynomial sequences are also explored.
