LC implies left nuclear square

This article gives the statement and possibly, proof, of an implication relation between two algebra loop properties. That is, it states that every algebra loop satisfying the first algebra loop property (i.e., LC-loop) must also satisfy the second algebra loop property (i.e., left nuclear square loop)
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Statement

Any LC-loop is a left nuclear square loop.

Definitions used

LC-loop

Further information: LC-loop

An algebra loop $(L, *)$ is termed a LC-loop if, for all $x, y, z \in L$ :

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

Left nuclear square loop

Further information: Left nuclear square loop

An algebra loop $(L, *)$ is termed a left nuclear square loop if every square element is in the left nucleus. In other words, for all $x, y, z \in L$ , we have:

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

Related facts

Facts used

LC implies left alternative: This states that any LC-loop is a left alternative loop, i.e., it satisfies the following universal identity:

$x * (x * y) = (x * x) * y$

Proof

Given: An algebra loop $(L, *)$ that is LC, i.e., for all $x, y, z \in L$ , we have: $(x * x) * (y * z) = (x * (x * y)) * z$

To prove: $L$ is a left nuclear square loop, i.e., for all $x, y, z \in L$ , we have: $(x * x) * (y * z) = ((x * x) * y) * z$

Proof: By the LC condition, we have, for all $x, y, z \in L$ , that:

$(x * x) * (y * z) = (x * (x * y)) * z (1)$

Further, by fact (1), $L$ is left alternative, so we obtain that:

$x * (x * y) = (x * x) * y (2)$

Plugging (2) into the right side of (1) gives:

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

as desired.