Division Algebras, Lie Algebras, Lie Groups: 10

Division Algebras, Lie Algebras, Lie Groups and Spinors

10. Octonion Associated Matrices and Adjoint Algebras

It's very easy to associate matrices with octonion units. I've done it with e₁ and e₂
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] = e₁X, the second matrix with the action e_L2[X] = e₂X, and the product
of the second matrix times the first with the action:
e_L12[X] = e₁(e₂X).
(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.