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**Division Algebras,
Lie Algebras, Lie Groups and Spinors**

This gif animation
illustrates the most popular of the cyclic octonion multiplication tables

(the dual of the one employed in the
book, but used by John Conway, Martin Cederwall

and Pertti Lounesto, among others, so I have changed my preference). As this
is covered

in detail elsewhere on this site, I won't go into too much detail here, except
to say that

e_{0}=1 is the octonion identity, and e_{a}, a=1,...,7, will
be refered to as the imaginary units.

Like we did for
the quaternions, if A is an imaginary octonion (no real part), then

A = uθ, where |u|=1, and |A| = θ,
so u is an element of the imaginary octonion

unit 6-sphere and is the direction of A, and θ
is the magnitude of A. Again, u^{2} = -1,

so u behaves like i when exponentiated, and exp(A) = exp(uθ)
= cos(θ) + u sin(θ).

The set of unit octonions = {exp(A): A imaginary} = S^{7} = 7-sphere.
Again, since

|XY| = |X||Y|, X and Y arbitrary octonions, this 7-sphere is closed under multiplication,

and it is easily shown to be parallelizable. But it is NOT a Lie group, because
it is not

associative, and hence it can NOT be represented by a matrix algebra.

But there are real
matrices associated with the octonions, and lots of Lie groups. But for

that we need a new page. (Note: I said these matrices are associated with the
octonions,

not that they represent the octonions. I've seen lots of papers claiming to
represent

the octonions by real matrices - or complex. Can't be done. There are associations

that can be made, but real and complex matrix algebras are necessarily associative,

hence whatever algebra results is not the octonion algebra - it's something
else.

Don't be fooled by substitutes - look for the real octonion label. Of course,
if you

adopt a multiplication other than conventional matrix multiplication, anything
can be done,

and I do this on page 17 of this section, but my motivation is pure, and the
result beautiful.)

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