Middle nuclear square loop

Symbol-free definition
An algebra loop is termed a middle nuclear square loop if every defining ingredient::square element is in the defining ingredient::middle nucleus.

Definition with symbols
An algebra loop $$(L,*)$$ is termed a middle nuclear square loop if it satisfies the following identity for all $$x,y,z \in L$$:

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