Left nuclear square loop

Definition
An algebra loop $$(L,*)$$ is termed a left nuclear square loop if every square element, i.e., every element of the form $$x * x$$, is in the defining ingredient::left nucleus. In other words, the following identity is satisfied for all $$x,y,z \in L$$:

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

Incomparable properties

 * Middle nuclear square loop
 * Right nuclear square loop