Split Octonions and the Rolling Ball

You can define complex numbers as pairs of real numbers with the product

(a, b) (c, d) = (a c - d \overset{―}{b}, \overset{―}{a} d + c b)

where

\overset{―}{a} = a

for real numbers.

Then you can define define conjugation for complex numbers by $\overset{―}{(a, b)} = (\overset{―}{a}, - b)$ Iterating this process, you can define quaternions as pairs of complex numbers...

... and octonions as pairs of quaternions!