simply polar elements in a ring An element $a$  in a ring $A$ with identity is said to be simply polar if there is $b$ for which  $a=aba$, with $ab=ba$. 
If in addition $b=bab$ then such an element $b$ is unique. 
The question is why $b$ commutes with everything that commutes with $a$.

Assume $ac=ca$.
Then $abca=abac=ac=ca=caba=acba$, hence
$abc=bac=bca = babca=bacba=abcab = acbab=acb$, i.e. $bc-cb$ is left annihilated by $a$ (and by symmetry  is also right annihilated).
Then 
$a(bc-cb+b)a = aba=a$ and $a(bc-cb+b) = ab=ba = (bc-cb+b)a$ and the uniqueness of $b$ implies $bc-cb+b=b$ and finally $bc=cb$.
