  A novel procedure for evaluating Wigner coupling coefficients and Racah
recoupling coefficients for U(4) in two group-subgroup chains is presented. The
canonical U(4)->U(3)->U(2)->U(1) coupling and recoupling coefficients are
applicable to any system that possesses U(4) symmetry, while the physical
U(4)->SU_S(2)xSU_T(2) coupling coefficients are more specific to nuclear
structure studies that utilize Wigner's Supermultiplet Symmetry concept. The
procedure that is proposed sidesteps the use of binomial coefficients and
alternating sum series, and consequently enables fast and accurate computation
of any and all U(4)-underpinned features. The inner multiplicity of a (S,T)
pair within a single U(4) irreducible representation is obtained from the
dimension of the null space of the SU(2) raising generators; while the
resolution for the outer multiplicity follows from the work of Alex et al. on
U(N). It is anticipated that a C++ library will ultimately be available for
determining generic coupling and recoupling coefficients associated with both
the \textit{canonical} and the \textit{physical} group-subgroup chains of U(4).
