  The matrix elements of the quadrupole collective variables, emerging from
collective nuclear models, are calculated in the natural Cartan-Weyl basis of
O(5) which is a subgroup of a covering $SU(1,1)\times O(5)$ structure. Making
use of an intermediate set method, explicit expressions of the matrix elements
are obtained in a pure algebraic way, fixing the $\gamma$-rotational structure
of collective quadrupole models.
