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.
Categories: math-ph hep-th math.MP