Let $K$ be a skew-field, infinite dimensional over its center $F$.

From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-algebras have finite dimension over their center).

There, a GPI (generalized polynomial identity) has coefficients from the center $F$. If the coefficients are arbitrary, one has a GI (generalized identity), and a theorem by Amitsur describing the possible structure of $K$.

In my research, I now came upon skew-fields which might satisfy a "skew" GPI in the following sense: Let $\sigma$ be an involutory antiautomorphism of $K$ and $\gamma$ an automorphism of $K$ of order 1 or 2. For simplicity, let's restrict to the case that $\sigma$ and $\gamma$ commute.

Is it possible that $K$ satisfy an "skew" GPI, meaning that coefficients are arbitrary from $K$, and the GPI contains not just $x$, but also $x^\gamma$, $x^\sigma$ and $x^{\gamma\sigma}$ ? The case with a single unknown is all that interests me.

I actually kinda hope these skew fields don't exist respectively must be finite dimensional over their center.