Unital magma of size two is abelian monoid

Statement
Suppose $$(S,*)$$ is a magma of size two and has a neutral element $$e$$. Then, $$*$$ is both a commutative binary operation and an associative binary operation, making $$(S,*)$$ an abelian monoid.

Facts used

 * 1) uses::Subset of elements in magma that commute with every element cannot have complement of size one