Hello all,
For all who might be interested...
I wanted to point out that expression of the octonion product
in terms of the Clifford product here for <Cl<sub>0,7</sub>><sub>0,1</sub> (the paravectors of Cl<sub>0,7</sub>) is not as complicated as it may look...
What happens is that the Clifford product produces scalars, 1-vectors, and bivectors... The '1' in (1 - v) catches the scalar and vector components of the Clifford product and passes them over to the octonion product unchanged. The components of the non-simple trivector v of R<sup>0,7</sup> catches all the bivector portions of the Clifford product and duals them into their appropriate octonion product 1-vectors.
Scalars multiplied by v produce trivectors... 1-vectors multiplied by v produce bivectors and 4-vectors... and all of these are excluded since they are not of grades 0 or 1.
In actual computations, only one of the simple trivectors of v will dual with any bivector generated by the Clifford product... so, when you do this by hand, it's not quite as involved as it might at first look... Since you know that only the 1, or one element of v will multiply and contribute the resulting octonion product from the input Clifford product on the left. And it's pretty obvious which term will be the appropriate one for each term of the Clifford product...
I don't know about any of you,... but, for me, v looked pretty scary at first. It took me awhile to see that this is all that Lounesto means for us to do.
Of course, I think it's still easier to figure out an octonion product by hand by using the heptagon and the simple rule illustrated in my post about seeking an alternate basis for the octonions.
Also... how about forming a basis for the quaternions in <Cl<sub>0,3</sub>><sub>0,1</sub> utilizing the product rule
...?
One could probably also adapt how Lounesto forms octonions and their product in <Cl<sub>0,8</sub>><sub>1</sub> similarly for forming quaternions and their product in <Cl<sub>0,4</sub>><sub>1</sub>...?
Likewise, one could also probably make a similar formulations for the complex numbers in <Cl<sub>0,1</sub>><sub>0,1</sub> and <Cl<sub>0,2</sub>><sub>1</sub>...?
What I think is neat is that it appears one can get all the normed division algebras (the reals are kinda a trivial case...) into one (or two) formulations...?
Another question... definitions. Are these two different types of representations the <i>Pinor</i> and <i>Spinor</i> representations? Is this what those terms mean?