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

It's very easy
to associate matrices with octonion units. I've done it with e_{1} and
e_{2}

above. There is a unique matrix derived in this way for each of the 8 basis
units

(including the identity). Like the imaginary octonion units themselves, the
7 matrices

associated with the imaginary octonion basis units anticommute, but unlike the

octonion units, these matrices do not close under multiplication. In fact, via
matrix

products and sums they generate all of **R**(8) = CL(0,6), a 64-dimensional
algebra.

However, although the product of the two matrices above may not be associated

with a single octonion unit, it is in fact associated with a more complicated
octonion

action arising out of octonion nonassociativity.

For X an arbitrary
octonion, the first matrix is associated with the action,

e_{L1}[X] = e_{1}X, the second matrix with the action e_{L2}[X]
= e_{2}X, and the product

of the second matrix times the first with the action:

e_{L12}[X] = e_{1}(e_{2}X).

(Ignore the fact that the matrix product is associated with the reverse octonion

product; what's important is that the product is in fact associated with this
embedded

action involving two octonion units; we won't be using the matrices after this.)

These ideas were
developed in fairly deep detail in the
book. We'll go into them

a little quicker here.

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