Central subloop

Definition
A subloop of an algebra loop is termed a central subloop or central subgroup if it is contained in the center of the loop. In other words, a subloop $$S$$ of an algebra loop $$(L,*)$$ if, for all $$a \in S$$ and $$x,y \in L$$, we have:

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

Note that any central subloop is an abelian group under the induced multiplication because the multiplication operation on it is associative.