Complete sets of commuting operators and O(3) scalars in the enveloping algebra of SU(3)

We consider the "missing label" problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3) ⊃ O(3) ⊃ O(2). We prove that only two independent O(3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L² and the two SU(3) Casimir operators C ⁽²⁾ and C⁽³⁾. Any one of these two operators (of third and fourth order in the generators) can be added to C⁽²⁾, C ⁽³⁾, L², and L₃ to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X⁽³⁾ and X⁽⁴⁾ are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.