Classification of some graded not necessarily associative division algebras I

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group $G$, and have a basis $\{v_g\mathrm{\mid }g\in G\}$ as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in $\{1,-1\}$. We classify here those graded by an abelian group $G$ of order $\mathrm{\mid }G\mathrm{\mid }\le 8$ with $G$ non–isomorphic to $\text{z}\mathrm{/}8\text{z}$. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.