Page 2.

Two Tables

There are 7 hypercomplex octonion units: e_a, a = 1,...,7, and we let
e₀ = 1, the identity. For a = 1,...,7 they satisfy

e_a² = -1.

This is a property they share with the complex algebra.
Like the quaternion algebra the hypercomplex units anticommute:
e_a e_b = -e_b e_a
for distinct a and b from 1 to 7. But unlike either the complexes or
quaternions, the octonions do not associate.

I'll look at the D(+) and C(+) multiplication tables here.
They are dual to each other, each sharing the same elegant properties.
In particular, in both cases, if one has

e_a e_b = ± e_c

(a,b,c integers from 1 to 7), then that immediately implies that

e_2a e_2b = ± e_2c

and

e_4a e_4b = ± e_4c

and

e_a+k e_b+k = ± e_c+k

(k an integer), where in all cases the indices are taken from 1 to 7, modulo 7.
Note, because of this modulo 7 property, e_8a = e_a,
so those first two properties hold in general for any power of 2 in the subscript,
but nothing new is gained after 2² = 4.

It is worth noting that these index doubling and index cycling
properties make it easy to generalize particular products and properties to the whole algebra.

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