Invertible elements of alternative unital ring form Moufang loop

Statement
Suppose $$R$$ is an alternative unital ring with multiplication $$\! *$$. Suppose $$S$$ is the subset of $$R$$ comprising those elements of $$R$$ that possess two-sided inverses for $$\! *$$. Then, $$S$$ is closed under $$*$$ and acquires the structure of a fact about::Moufang loop under $$*$$.

Related facts

 * Artin's theorem on alternative rings which states that for a ring, being alternative is the same as being diassociative.
 * Alternative ring satisfies Moufang identities

Facts used

 * 1) uses::Alternative ring satisfies Moufang identities

Proof
The proof follows directly from fact (1).