  We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory that
approach BPS bounds. These limits result in non-relativistic 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. In previous works we have considered
various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current
work, we will construct the remaining Spin Matrix Theories defined near the
$\frac{1}{8}$-BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases.
We derive the Hamiltonians by applying the spherical reduction algorithm and
show that they match with the spin chain result, coming from the loop
corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the
positivity of the spectrum by constructing cubic supercharges using the
enhanced PSU$(1|1)^2$ symmetry and show that they close to the interacting
Hamiltonian. We finally analyse the symmetry structure of the sectors in view
of an interpretation of the interactions in terms of fundamental blocks.
