Proposition 4

Let $E$ be an associative magma, $X$ and $Y$ two subsets of $E$. If every element of $X$ commutes with every element of $Y$ every element of the stable subset generated by $X$ commutes with every element of the stable subset generated by $Y$.

Let $X'$ and $X''$ be the centralizer and bicentralizer of $X$. They are stable subsets of $E$. Now $X \subset X''$ and $Y \subset X'$ and hence $X''$ (resp. $X'$) contains the stable subset of $E$ generated by $X$ (resp. $Y$). As every element of $X''$ commutes with every element of $X'$, the proposition follows.