  We obtain many objects of discrete differential geometry as reductions of
skew parallelogram nets, a system of lattice equations that may be formulated
for any unit associative algebra. The Lax representation is linear in the
spectral parameter, and paths in the lattice give rise to polynomial
dependencies. We prove that generic polynomials in complex two by two matrices
factorize, implying that skew parallelogram nets encompass all systems with
such a polynomial representation. We demonstrate factorization in the context
of discrete curves by constructing pairs of B\"acklund transformations that
induce Euclidean motions on discrete elastic rods. More generally, we define a
hierarchy of discrete curves by requiring such an invariance after an integer
number of B\"acklund transformations. Moreover, we provide the factorization
explicitly for discrete constant curvature surfaces and reveal that they are
slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method,
this interpretation constructs such surfaces from given discrete holomorphic
maps.
