Ciro Santilli 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. with complex number multiplication
- with the cross product
- quaternions, i.e. with the quaternion multiplication
An algebra over a field where division exists.
Notably, the octonions are not associative.