What kind of algebra and how can I learn its properties

Question

I try to find any information on the following algebra defined as: $$(a,b)+(c,d)=(a+c,b+d)$$ $$(a,b)*(c,d)=(ac,ac-bd)$$ It is non-associative, but commutative and distributive. Can it be classified somehow? Do there exist any information on it? How can I find its properties?

Answer 1

Sounds like you're maybe talking about general non-associative algebras (also known as distributive algebras). Commutativity is just then a property that non-associative algebras can have or not have. I don't always trust Wikipedia but this page would be a good start. There are some good examples like Jordan Algebras