  The purpose of this study is to extend the concept of a generalized Lie $3-$
algebra, known to the divisional algebra of the octonions $\mathbb{O}$, to
split-octonions $\mathbb{SO}$, which is non-divisional. This is achieved
through the unification of the product of both of the algebras in a single
operation. Accordingly, a notational device is introduced to unify the product
of both algebras. We verify that $\mathbb{SO}$ is a Malcev algebra and we
recalculate known relations for the structure constants in terms of the
introduced structure tensor. Finally we construct the manifestly
super-symmetric $\mathcal{N}=1\;\mathbb{SO}$ affine super-algebra. An
application of the split Lie $3-$algebra for a Bagger and Lambert gauge theory
is also discussed.
