##
**Octonions**

*
The following brief introduction to the octonions is adapted from any of
several papers
*

which can be found
here,
and which include further references to the literature.

The **octonions O** are the nonassociative, noncommutative, normed
division algebra over the real numbers. They can be expressed in terms of a
natural basis, which we denote **i,j,k,kl,jl,il,l**, together with
the identity element, 1. Each basis element except 1 squares to -1, that is
**i**^{2}=**j**^{2}=-1, and so forth.
The octonions are thus a generalization of the complex numbers, with 7
imaginary units rather than just one.

The full multiplication table is conveniently encoded in the 7-point
projective plane, shown above. The product of any two imaginary units is
given by the third unit on the unique line connecting them, with the sign
determined by the relative orientation. For instance,
**i j**=**k**, **i l**=**il**, as well as
cyclic permutations of these relations, but **j i**=**-k**.

The *associator* of three octonions is
[**a**,**b**,**c**]
*=(***ab**)**c** - **a**(**bc**)
which is totally antisymmetric in its arguments and has no real part.
Although the associator does not vanish in general, the octonions do satisfy a
weak form of associativity known as *alternativity*, namely
[**a**,**b**,**a**]*=*0.
(The underlying reason for this is that two octonions determine a quaternionic
subalgebra of the octonions, so that any product containing only two
octonionic directions is associative.)

*Octonionic conjugation* is given by reversing the sign of the imaginary
basis units. Conjugation is an antiautomorphism, since it satisfies
*(***ab**)^{*} = **b**^{*} **a**^{*}.
The *inner product* on **O** is the one inherited from
**R**^{8}, which can be rewritten
**a**·**b**
= (**ab**^{*} + **ba**^{*})/2
= (**b**^{*}**a** + **a**^{*}**b**)/2.
Finally, the *norm* of an octonion is just
|**a**|^{2} = **aa**^{*}
which satisfies the defining property of a normed division algebra, namely
|**ab**| *=* |**a**| |**b**|

tevian@math.orst.edu