Left nuclear square magma

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.
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Definition

Symbol-free definition

A magma is termed a left nuclear square magma if every square element of the magma is inside the left nucleus.

Definition with symbols

A magma $(S,*)$ is termed a left nuclear square magma if the following identity is satisfied for all $x,y,z\in S$:

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

An algebra loop that is also a left nuclear square magma is termed a left nuclear square loop.