Skew-commutative implies flexible

Statement
Any skew-commutative ring is a flexible ring.

Skew-commutative ring
A non-associative ring $$R$$ with addition $$+$$ and multiplication $$*$$ is termed skew-commutative if, for all $$x,y \in R$$, we have:

$$\! (x * y) + (y * x) = 0$$

Flexible ring
A non-associative ring $$R$$ with multiplication $$*$$ is termed flexible if, for all $$x,y \in R$$, we have:

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