  Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras
with four colors, four bosons, and four fermions have been computed. In this
paper, we further analyze these results in terms of $BC_3$, the signed
permutation group of three elements, and $BC_4$, the signed permutation group
of four elements. It is shown how all 36,864 adinkras can be generated via
$BC_4$ boson $\times$ $BC_3$ color transformations of two quaternion adinkras
that satisfy the quaternion algebra. An adinkra inner product has been used for
some time, known as the \emph{gadget}, which is used to distinguish adinkras.
We~show how 96 equivalence classes of adinkras that are based on the gadget
emerge in terms of $BC_3$ and $BC_4$. We also comment on the importance of the
gadget as it relates to separating out dynamics in terms of K\"ahler-like
potentials. Thus, on the basis of the complete analysis of the supersymmetrical
representations achieved in the preparatory first four sections, the final
comprehensive achievement of this work is the construction of the universal
$BC_4$ non-linear $\sigma$-model.
