The class of Jordan algebras are the most important class of algebras in this direction.

They are defined by the two identities,

(commutativity): $xy=yx$,

(Jordan identity): $(xy)(xx)=x(y(xx))$.

They are an extremely important class of non-associative algebras (check Jacobson's Strucuture and Representation of Jordan algebras or McCrimmon's A taste of Jordan algebras)

The most easy way to produce such an algebra is by considering an associative (alternative suffices) algebra $A$ and changing the product from $.$ to $*$ as follows: $a*b = \frac{ab+ba}{2}$. Some care of course is necessary in case of charestic two.

The class of Jordan algebras are the most important class of algebras in this direction.

They are defined by the two identities,

(commutativity): $xy=yx$,

(Jordan identity): $(xy)(xx)=x(y(xx))$.

They are an extremely important class of non-associative algebras (check Jacobson's Strucuture and Representation of Jordan algebras or McCrimmon's A taste of Jordan algebras)

The most easy way to produce such an algebra is by considering an associative (alternative suffices) algebra $A$ and changing the product from $.$ to $*$ as follows: $a*b = \frac{ab+ba}{2}$. Some care of course is necessary in case of characteristic two.