  We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory that
approach BPS bounds. These limits result in non-relativistic near-BPS theories
that describe the effective dynamics near the BPS bounds and upon quantization
are known as Spin Matrix theories. The near-BPS theories can be obtained by
reducing $\mathcal{N}=4$ SYM on a three-sphere and integrating out the fields
that become non-dynamical in the limits. We perform the sphere reduction for
the near-BPS limit with $\mathrm{SU}(1,2|2)$ symmetry, which has several new
features compared to the previously considered cases with $\mathrm{SU}(1,1)$
symmetry, including a dynamical gauge field. We discover a new structure in the
classical limit of the interaction term. We show that the interaction term is
built from certain blocks that comprise an irreducible representation of the
$\mathrm{SU}(1,2|2)$ algebra. Moreover, the full interaction term can be
interpreted as a norm in the linear space of this representation, explaining
its features including the positive definiteness. This means one can think of
the interaction term as a distance squared from saturating the BPS bound. The
$\mathrm{SU}(1,1|1)$ near-BPS theory, and its subcases, is seen to inherit
these features. These observations point to a way to solve the strong coupling
dynamics of these near-BPS theories.
