SU(3) symmetry for orbital angular momentum and method of extremal projection operators

Basic elements of the formalism of the theory that is based on the representations of the SU(3) group for the case of its reduction to the SO(3) subgroup of orbital angular momentum and which is widely used in theoretical physics is presented in a systematic and consistent form. Irreducible SU(3) ? SO(3) bases, both a nonorthogonal one chosen among Elliott vectors and an orthogonal one obtained from a nonorthogonal one by diagonalizing the Bargmann-Moshinsky operator, are described in detail. In particular, it is shown that there is wide arbitrariness in choosing a basis among Elliott vectors. The SU(3) ? SO(3) Clebsch-Gordan coefficients are considered in detail, along with all of their classical symmetry properties. A brief survey (history of discovery) of the method of extremal projection operators for Lie symmetries (Lie algebras and superalgebras and their quantum analogs) is given.