Right-associative elements of loop form subgroup

Statement
Suppose $$(L,*)$$ is an algebra loop. Then, the set of right-associative elements of $$L$$ is nonempty and forms a subgroup of $$L$$. This subgroup is termed the right kernel of $$L$$ or the right-associative center of $$L$$.

Right-associative element
An element $$c \in L$$ is termed right-associative if, for all $$a,b \in L$$, we have:

$$(a * b) * c = a * (b * c)$$.

Related facts

 * Left-associative elements of loop form subgroup: Note that the proofs are identical and the facts can be deduced from each other using the opposite magma construction.

Facts used

 * 1) uses::Right-associative elements of magma form submagma
 * 2) uses::Monoid where every element is left-invertible equals group

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

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

Proof: