Conjugate Real Classes in General Linear Groups

Let $\text{F}$ be a field with a non-trivial involution $c:\alpha \to {\alpha }^c$. An element $g\in {\mathrm{G}\mathrm{L}}_n(\text{F})$ is called $c$-real if it is conjugate to $(g^c{)}^{-1}$. We prove that for $n\ge 2$, $g\in {\mathrm{G}\mathrm{L}}_n(\text{F})$ is $c$-real if and only if it has a representation in some unitary group of degree $n$ over $\text{F}$.