[JadeNB]

> The spinors/vectors relevant to the article are those of Spin(8), which has something to do with triality (still need to understand all of this better myself).

Spin(8) itself doesn't have much to do with triality; it's just that triality describes an unusual symmetry among representations of Spin(8), due to an unusual outer automorphism. (Of course, from some perspectives, that means that Spin(8) has everything to do with triality, but I hope my meaning will be clear.) The best accessible mathematical explanation of triality I know is from Baez: https://math.ucr.edu/home/baez/octonions/node7.html.

> So a spinor could be said to be anything whose symmetries are a Spin group. Spin groups are easily constructed in clifford algebras and it turns out that they have matrix representations. Whenever you have matrices (linear maps) you may wonder what the vector space is that they act on (i'm now using the term "vector" abstractly, not in contrast to spinors as above). Well, those are the spinors (technically pinors)!

One has to be a little careful here, because algebras have lots of representations, and there's no one representation that a priori may be said to be "the vector space on which they act" ("the" rather than "a"). For example—though it's a bad example because it's not a faithful action—Spin(8) naturally double covers SO(8), but we don't want to take the resulting 8-dimensional orthogonal representation (the "vector representation"). Instead, we want to take one of the three fundamental representations permuted by the triality automorphism (the V_1, V_2, V_3 in Baez's article).

[aap_]

I slightly edited my post while you were writing yours. I'm aware of the article but I still think it could be clearer. I'm a big fan of constructive proofs but that does not seem the case for most mathematicians, so i find a lot of stuff in math very unsatisfying and have to rediscover it myself via different routes. Educational for sure but not always so easy.

And yes, of course there are many possible matrix reps, sorry i was not being precise. The ones i was referring to are the ones given here: https://en.wikipedia.org/wiki/Classification_of_Clifford_alg...