Wondering if there are constructions related to the Cayley-Dickson construction in which it is built not out of real numbers but of arbitrary abstract objects. Similar to how you can have polynomial rings that aren't of the real numbers, or algebras over things other than real numbers. Basically, if you can use the Cayley-Dickson construction without real-numbers, with, say, vectors of arbitrary dimension. So it would go:
- 1-dimension: 3-variable vector instead of real number.
- 2-dimension: 2 3-variable vectors instead of complex number.
- etc.