More on the isomorphism SU(2) ⊗ SU (2) ≅ SO(4)

In this paper, we revisit the isomorphism SU(2) circle times SU( 2) congruent to SO(4) to apply to some subjects in Quantum Computation and Mathematical Physics. The unitary matrix Q by Makhlin giving the isomorphism as an adjoint action is studied and generalized from a different point of view. Some problems are also presented. In particular, the homogeneous manifold SU(2n)/SO( 2n) which characterizes entanglements in the case of n = 2 is studied, and a clear-cut calculation of the universal Yang-Mills action in (hep-th/0602204) is given for the abelian case.