Abstract
We develop the basic theory of Octonions, including various identities
and properties of the octonions and of the octonionic product, a
description of 7D isometries and representations of orthogonal
transformations. To this end we first develop the theory of the vector
cross product in 7 dimensions. The development of the theory of
Octonions is inspired by that of the theory of Quaternions by Lawrence
Paulson. However, we do not work within the type class real_algebra_1
because the octonionic product is not associative.