We give an explicit characterization of the most general quasi-Hermitian operator H, the associated metric operators eta+, and eta+- pseudo-Hermitian operators acting in C-2. The latter represent the physical observables of a model whose Hamiltonian and Hilbert space are respectively H and C-2 endowed with the inner product defined by eta+. Our calculations allows for a direct demonstration of the fact that the choice of an irreducible family of observables fixes the metric operator up to a multiplicative factor.