What's written below is a proof and discussion of the following:

Let g

g

Then for all distinct a,b = 1,...,7, there exists a unique A

the g

g

Define

g

This is an octonion product.

A related result, which will not be proved, is the following:

Let g_{a}, a=1,...,7, and X be as above, and let U be a linear in the g_{a}

and I (U is assumed nonzero and is necessarily invertible). Then for all distinct

a,b = 1,...,7, there exists a unique B_{UXab}linear in the g_{a}and I such that

g_{a}U g_{b}X = B_{UXab}X.

Define

g_{a}©_{ux}g_{b}= B_{UXab}.

This is an octonion product with U^{-1}the product identity (this is

a matrix way of creating an X,Y-product).

For A and X arbitrary octonions define the

A

Further define

e

e

It is not hard to prove that the span of the set of all such maps is isomorphic

to the

In particular, the distinct elements of the form

e

e

e

e

where a,b,c = 1,...,7, form a basis for the algebra of left adjoint actions

of

Well,

of the Clifford algebra

with negative definite metric. We can make the following multivector basis assignments:

e

e

e

e

e

e

e

The spinor space of

8 real numbers. The spinor space of

space

relationship to Lie groups like SL(2,O) come from trying to do Lie group things with

instead of

nonassociativity makes it an inappropriate choice for trying to generate associative algebras;

this is the message of the mathematics;

(equality - not just isomorphism)) is an inherent part of the mathematics of

Now that we've established the roles everything is to play, consider the subspace of

spanned by the elements e

this subspace is isomorphic to so(8), the Lie algebra of the orthogonal group SO(8).

The subspace spanned by the elements e

(it is assumed a and b are distinct), generates an so(7) Lie subalgebra of so(8).

Let X be an arbitrary nonzero element of

(if X is a unit octonion then this is just the conjugate). Note:

e

(e

c = 1,...,7; and it is unique, given X, e

(linear in the e

linear in the e

A[X] = A

This gives us a way of deriving multiplication tables for

g

which also anticommutes with the g

g

The set of distinct g

Choose a nonzero spinor X (a column of 8 real numbers). For all distinct a,b = 1,...,7, there exists

a unique A

g

For all distinct a,b define the product

g

It should now be somewhat clear that the set of g

the o

(extend the basis to the identity, and the product in the obvious ways, to complete the picture).

Analogous but simplified reasoning on

I don't believe it would be difficult to prove that this trick would not work on

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