where { Ŝi , Ŝj } + represents the anti-commutator between Ŝi and Ŝj , the indices i, j, k ∈ {α nm , β nm , γ n }, and f ijk and d ijk are the totally anti-symmetric and totally symmetric structure constants, respectively. Using Eqs. 8a-8b, one can obtain the following well-known expressions for these structure constants