1. The algebra of bioctonions is an example of an octonion algebra. 2. But like the octonion product it is not uniquely defined. 3. In particular, this gives the classification of octonion algebras. 4. Moreover, the Moufang identities hold in any octonion algebra. 5. The Cayley plane uses octonion coordinates which do not satisfy the associative law. 6. There are corresponding split octonion algebras over any field " F ". 7. Instead there are many different cross products, each one dependent on the choice of octonion product. 8. This property is shared by most binary operations, but not subtraction or division or octonion multiplication. 9. This one is derived from its parent octonion ( one of 480 possible ), which is defined by: 10. The algebra of bioctonions is the octonion algebra over the complex numbers "'C " '.
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