Connection between two Wigner functions for spin systems

In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU(2) that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension;of the group and serves for all square-integrable representations; for the SO(3) case this entails a three-dimensional "metaphase" space; Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.