  In this work, we analyze two models beyond the Standard Models descriptions
that make ad hoc hypotheses of three point-like lepton and quark generations
without explanations of their physical origins. Instead of using the same Dirac
equation involving four anti-commutative matrices for all such structure-less
elementary particles, we consider in the first model the use of sixteen
direct-product matrices of quaternions that are related to Diracs gamma
matrices. This associative direct-product matrix model could not generate three
fermion generations satisfying Einsteins mass-energy relation. We show that
sedenion algebra contains five distinct quaternion sub-algebras and three
octonion sub-algebras but with a common intersecting quaternion algebra. This
model naturally leads to precisely three generations as each of the
non-associative octonion sub-algebra leads to one fermion generation. Moreover,
we demonstrate the use of basic sedenion.
