From your post I can tell you're way better at math/geometry than me, but I understood 80% of that. :)
Cayley-Dickson is interesting especially for Physics of course, because it brings in the concept of 'variable dimensions'. I think the flattening of objects, and the stopping of clocks (in Relativity), due to Lorentz effects in Minkowski space both on Black Hole Event Horizons and for objects approaching light speed (anywhere Lorentz holds) is, at the limits, ultimately the loss of a dimension, which would be my overall interpretation of what Cayley-Dickson is about too, in very broad terms.
So if Minkowski space is 4 dimensional, there would be some geometry for a 5-Dim Minkowski and it would use Octonians maybe, and that would be the geometry of the universe our universe is "embedded in"...I mean assuming of course you believe our universe is a Black Hole and we are all on an Event Horizon embedded in a 5D universe. Ya know, as one does. lol.
Cayley-Dickson is interesting especially for Physics of course, because it brings in the concept of 'variable dimensions'. I think the flattening of objects, and the stopping of clocks (in Relativity), due to Lorentz effects in Minkowski space both on Black Hole Event Horizons and for objects approaching light speed (anywhere Lorentz holds) is, at the limits, ultimately the loss of a dimension, which would be my overall interpretation of what Cayley-Dickson is about too, in very broad terms.
So if Minkowski space is 4 dimensional, there would be some geometry for a 5-Dim Minkowski and it would use Octonians maybe, and that would be the geometry of the universe our universe is "embedded in"...I mean assuming of course you believe our universe is a Black Hole and we are all on an Event Horizon embedded in a 5D universe. Ya know, as one does. lol.