Ciro Santilli @cirosantilli 37

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

Constructs the quaternions from complex numbers, octonions from quaternions, and keeps doubling like this indefinitely.

There are 3: real numbers, complex numbers and quaternions.

Notably, the octonions are not associative.

Unlike the quaternions, it is non-associative.

Double cover of $SO(3)$.

Isomorphic to the quaternions.