A split octonion algebra is one for which the quadratic form N is isotropic (i.e., there exists a non-zero vector x with N(x) = 0).
The construction of general octonion algebras over an arbitrary field k was described by Leonard Dickson in his book Algebren und ihre Zahlentheorie (1927) (Seite 264) and repeated by Max Zorn.
[2] The product depends on selection of a γ from k. Given q and Q from a quaternion algebra over k, the octonion is written q + Qe.
Cohl Furey has proposed that octonion algebras can be utilized in an attempt to reconcile components of the Standard Model.
Moreover, the possible norm forms are exactly the Pfister 3-forms over F.[4] Since any two octonion F-algebras become isomorphic over the algebraic closure of F, one can apply the ideas of non-abelian Galois cohomology.