Commutative implies flexible

Statement
Suppose $$(S,*)$$ is a commutative magma, i.e, we have:

$$a * b = b * a \ \forall \ a,b \in S$$

Then, $$(S,*)$$ is also a flexible magma, i.e., we have:

$$x * (y * x) = (x * y) * x \ \forall \ x,y \in S$$

This also shows that a commutative non-associative ring must be a fact about::flexible ring.

Proof
Given: A magma $$(S,*)$$ where $$a * b = b * a$$ for all $$a,b \in S$$.

To prove: $$x * (y * x) = (x * y) * x$$ for all $$x,y \in S$$.

Proof:

By commutativity between $$x$$ and $$y$$, we have:

$$\! x * (y * x) = x * (x * y) \ \qquad \ (\dagger)$$

Further, by commutativity of $$x$$ and $$x * y$$, we have:

$$\! x * (x * y) = (X * y) * x \ \qquad \ (\dagger\dagger)$$

Combining $$(\dagger)$$ and $$(\dagger\dagger)$$, we obtain what we need to prove.