Algebra over a field

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

An algebra over a field where division exists.

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

Notably, the octonions are not associative.