Quadrupole collective variables in the natural Cartan–Weyl basis

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) × 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 γ-rotational structure of collective quadrupole models.