Left-associative elements of loop form subgroup

Statement
Suppose $$(L,*)$$ is a loop. Then, the left nucleus of $$L$$, i.e., the set of left-associative elements of $$L$$, is nonempty and forms a subgroup of $$L$$. This subgroup is sometimes termed the left kernel of $$L$$ or the left-associative center of $$L$$.

Related facts

 * Right-associative elements of loop form subgroup

Facts used

 * 1) uses::Left-associative elements of magma form submagma: Further, this submagma is a subsemigroup, and if the whole magma has a neutral element, it has the same neutral element and becomes a monoid.
 * 2) uses::Monoid where every element is right-invertible equals group, which in turn uses uses::equality of left and right inverses in monoid

Proof
Given: A loop $$(L,*)$$ with identity element $$e$$. $$S$$ is the set of left-associative elements of $$L$$.

To prove: $$S$$ is a subgroup of $$L$$. More explicitly, $$(S,*)$$ is a group with identity element $$e$$.

Proof: