  The simplest supersymmetry (SUSY) algebra in four dimensional Euclidean space
($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four
dimensional Minkowski space ($4dM$). The structure of the former algebra is
examined in greater detail in this paper. We first present its Clifford algebra
structure. This algebra shows that the momentum Casimir invariant of physical
states has an upper bound which is fixed by the central charges. Secondly, we
use reduction of the $N = 1$ SUSY algebra in six dimensional Minkowski space
($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this
same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces
Zumino's SSYM in $4dE$. We demonstrate how this dimensional reduction can be
used to introduce additional generators into the SUSY algebra in $4dE$.
