I have a division ring $D$, $A$ a ring of endomorphisms of the additive group of $D$ and define the set and $G$ a subring of $A$ containing the right multiplications by elements of $D$. I define $E$ as follows.
$$E := \{d\in D\ \vert\ g\circ d^l = d^l\circ g\ \forall\ g\in G\}$$
The question is how can I show that $E$ is a division ring? The only point I'm missing is to show that if $d\in E$, then $d^{-1}\in E$.
$g(x)d^l=g(xd^l)$ so that $$ g(x)=g(xd^l)d^{-l} $$
If $x=yd^{-l}$, then $$ g(yd^{-l})=g(y)d^{-l} $$