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

- complex numbers, i.e. $R_{2}$ with complex number multiplication
- $R_{3}$ with the cross product
- quaternions, i.e. $R_{4}$ with the quaternion multiplication

An algebra over a field where division exists.

Notably, the octonions are not associative.