The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication with.
12.
A convenient mnemonic is given by the diagram at the right which represents the multiplication table for the split octonion.
13.
Up to isomorphism, the octonions and the split-octonions are the only two octonion algebras over the real numbers.
14.
Up to "'R "'- algebra isomorphism, these are the only octonion algebras over the reals.
15.
It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm.
16.
The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions.
17.
This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions.
18.
Conversely, an algebra with a non-degenerate symmetric bilinear form satisfying this equation is either a para-Hurwitz algebra or an eight-dimensional pseudo-octonion algebra.
19.
An algebraic symmetry is lost with each increase in dimensionality : quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.
20.
Note that in keeping associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for ".
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