There are 7 hypercomplex octonion units: ea, a = 1,...,7, and we let
e0 = 1, the identity. For a = 1,...,7 they satisfy
ea2 = -1.
This is a property they share with the complex algebra.
Like the quaternion algebra the hypercomplex units anticommute:
ea eb = -eb ea
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
ea eb = ± ec
(a,b,c integers from 1 to 7), then that
immediately implies that
e2a e2b = ± e2c
e4a e4b = ± e4c
ea+k eb+k = ± ec+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, e8a = ea,
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.
Octoshop III page.