A Coefficient Calculus for Unitary (g,K) Modules of SU(2,2)

We develop an explicit operator and coefficient calculus for admissible (g,K) modules of SU(2,2), where g=sl(4,C) and K=S(U(2)×U(2)). The K-types are indexed by triples (n,k,m). We introduce operators Aδ and Bδ associated with the noncompact roots, together with auxiliary operators Q and R, and establish their commutator relations, central coefficient relations, and, in the unitary setting, adjoint and norm identities. For multiplicity-one K-types, these results reduce part of the analysis to scalar coefficients arising from compositions such as AδBδ and BδAδ. We derive explicit coefficient relations and necessary sign restrictions and analyze families according to the minimal value N of n+m. For N=0, the coefficient system has a two-parameter structure and exhibits natural reducibility boundaries when certain coefficients vanish. For N>0, the method yields both larger K-type patterns and multiplicity-free ladder-type patterns. The resulting K-type patterns agree with those occurring in the known classification of the unitary dual of SU(2,2), although a complete comparison, including reducibility phenomena, is beyond the scope of the present paper. The coefficient calculus provides an explicit framework for studying unitary (g,K) modules through their K-type structure.