Please see the STA Wikipedia page or Lasenby, A. (2022), *Some recent results for SU(3) and Octonions within the Geometric Algebra approach to the fundamental forces of nature*, doi:10.1002/mma.8934 for more information on STA based octonion products.

Space-Time Algebra (STA) octonionic representation

I have updated my comprehensive Fano plane list of 480 octonion triad sets. This new file adds the SpaceTime Algebra (STA) octonion representation on the nodes here. It is a large PDF at 80Mb. It is sorted by the E8 root vertex numbers based on a canonical sort from the 9th row of the Pascal triangle. As such, it has 16 excluded roots that are the 8 positive and 8 negative generator roots (which don’t produce unique octonions).

There are now also two sets of files that show the STA detail data in each 240 + 240 sets of E8 indices splitting of the 480 (one with the first triad that defines the non-commutative multiplication by either Right (IJ=K) & Left (JI=K) quaternions. These are also 80Mb files. There are corresponding Mathematica notebooks with the same content in case you want to interact (e.g. rotate in 3D the Fano plane). They are here and here (respectively).

My original Fano plane / split Fano plane posts are here and here.

A bit about octonions:

The 480 octonion permutations are generated with a list of 30 sets of triples (3 numbers from 1-7 with no two triples with two numbers in common).

These can be visualized as a Fano plane. The Fano plane is easily constructed by taking the first 3 (of 7) triads and deleting the 4th and 7th (of the 9) numbers. This gives a “flattened” list of numbers (1-7) in some permutation order. While there are 7!=5040 possible, only 480 create proper octonions. But how to pair up the 480 to mesh with E8?

There are 16 possible +/- sign reversal (or sign mask (sm) bits also shown as hex number) permutations available in each of the 30 Fano planes (30*16=480). These sm bits also “invert” like the “anti” bits in E8 (which sorts it into a top half/bottom half canonical list built using the 9th row of the Pascal triangle and Clifford algebras). So the 16 sm possibilities split into two sets of 8. This changes the direction of the Fano plane arrows.

The algorithm to produce the 480 triad sets from the above sign masks and 30 canonical sets of 7 triads comes from a discussion by D. Chesley on twisted octonions and a paper “Octonion Multiplication and 7-Cube Vertices” from his personal website (1988).

In it he says: “Each representation of the octonions arranges the seven imaginary unit elements into seven triads, with no two distinct triads having a pair of elements in common. There is a natural mapping between the 30 distinct sets of triads and the 30 distinct versions of B7 which contain the vertex (0,0,0,0,0,0,0). Assigning a chirality to each triad of a given octonion representation then reveals that not all 128 possible chirality assignments result in a normed division algebra. In fact, for each of the 30 possible ways of grouping the elements, each of 16 chirality assignments (again corresponding to a B7) gives a distinct representation of the octonions for a total of 480 representations, and each of the other 112 (corresponding to D7) gives a distinct representation of the twisted octonion algebra.”

It turns out with some skillfull pattern matching in constructing the sign masks, the 8 sm bits pair up into two sets of four {1,4,6,7} and {2,3,5,8} giving a kind of “spin” match to E8. These pairs of 4 also operate as a left-right symmetry pattern seen in the structure of E8.