  Dwyer, Miller and Wilkerson proved that at the prime 2, the classifying
spaces of SU(2) and SO(3) can be obtained as a homotopy pushout of the
classifying spaces of certain subgroups. In this paper we show explicitly how
these decompositions arise from the fusion systems of SU(2) and SO(3) over
maximal discrete 2-toral subgroups.
