Left Bol implies left alternative

Statement for algebra loops
Any algebra loop that is a left Bol loop is also a left alternative loop.

Statement for magmas
A magma that is a fact about::left Bol magma and has a fact about::neutral element must be a fact about::left alternative magma.

Left Bol magma and loop
A magma $$(L,*)$$ is termed a left Bol magma if the following is satisfied for all $$x,y,z \in L$$:

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

A left Bol loop is a left Bol magma that is also an algebra loop.

Neutral element
A neutral element in a magma $$(L,*)$$ is an element $$e$$ such that $$x * e = e * x = x$$ for all $$x \in L$$. Note that any algebra loop has a neutral element by definition.

Left alternative magma and loop
A magma $$(L,*)$$ is termed a left alternative magma if the following is satisfied for all $$x,y \in L$$:

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

A left alternative loop is an algebra loop that is also a left alternative magma.

Proof
We prove the statement for magmas. Since an algebra loop has a neutral element by definition, the statement for algebra loops is an immediate consequence.

Given: A magma $$(L,*)$$ having a neutral element $$e$$, (i.e., $$x * e = e * x = x$$ for all $$x \in L$$) and satisfying the left Bol identity for all $$x,y,z \in L$$:

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

To prove: $$L$$ is left alternative, i.e., $$x * (x * y) = (x * x) * y \ \forall \ x,y \in L$$

Proof: Set $$y = e$$ in the left Bol identity. We get:

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

This simplifies to:

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

Since $$x,z$$ are universally quantified, we can replace the dummy variable $$z$$ by the dummy variable $$y$$, and obtain:

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