For some while, I have been intrigued but baffled by this mysterious topic.
One obstruction is that standard treatments all insist on dragging in the
notion of "spinors", which even Michael Atiyah admits to difficulty in
Another is that they gleefully wave hands over all manner of beautiful
properties, while steadfastly avoiding any discussion of how they might
algorithmically be realised.
An isolated honourable exception is provided by
Pertti Lounesto "Octonions and Triality"
Advances in Applied Clifford Algebras vol. 11 no.2, 191--213 (2001)
However my earlier attempt to implement his Cl(8) formulation failed,
through misinterpreting his notation. This difficulty has now been resolved,
and the latest upload at
includes a demo of octonions and triality in Spin(8) . [Inter alia, I have
at last succeeded in discovering an application for GA with grade exceeding
All later demos there in GA_scripts.py obviously require an accompanying
gloss summarising their motivation and methods: if somebody cares to
complain sufficiently vociferously, they might eventually receive one.
complication is my inability to appreciate why spinors (and representations in
general) might be useful or interesting at all: in particular I utterly fail
to grasp the purported connection between them and bosons and fermions of the
standard model in particle physics (of which my ignorance remains lamentably
Any available enlightenment would be appreciated!