  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 $\{v_g|g\in G\}$ 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 $|G|\leq 8$ with $G$ non--isomorphic to $\z/8\z$. We
will find the complex, quaternion, and octonion algebras, but also a remarkable
set of novel non--associative division algebras.
