aapI think what the Cayley-Dickson construction is doing at least up to the #octonions is "teaching you" how to move stereographically (SGly) in a space you already know. this starts from R to C as you SGly move along the line R. but this can also be unprojected onto a circle, so alternatively you "learn" how to rotate in 2D *instead*, C. Next step up is rotation and SG motion in 2D, which can be unprojected to rotation in 3D, H. then rotation and SG motion in 3D and that's where it stops: O
I call this stereographic motion ☯ rotation. It rotates between the origin and ∞. Or north and south pole if you want. But it does not pick a preferred direction of space for that, just in and out, ☯
R: just a point
C: (1D ☯ motion) xor 2D rotation
H: (2D rotation and ☯ motion) xor 3D rotation
O: 3D rotation and ☯ motion
This is even reminiscent of bott periodicity: at O you can't step up higher and still have ☯. it looks like you take O as the "interior" of a new point and loop back to R