To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the automorphism group of $G$ is generated by inner automorphisms and $\psi(x) = x (x^{-1})^t.$ For inner $\phi,$ we are asking which $x$ satisfy the quadratic equation $x = y a y a^{-}]$ for a fixed a (if $a$ is allowed to vary, it is well-known that every element of a complex semi-simple lie group is a commutator, so that should presumably imply that every element has that form for some $a$ ). For $\psi,$ we want to characterize matrices of the form $y = x (x^{-1})^t$ (by dimension counting this is a proper subset; it is pretty clear that it contains the complex orthogonal group).

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the automorphism group of $G$ is generated by inner automorphisms and $\psi(x) = (x^{-1})^t.$ For inner $\phi,$ we are asking which $x$ satisfy the quadratic equation $x = y a y a^{-1}$ for a fixed a (if $a$ is allowed to vary, it is well-known that every element of a complex semi-simple lie group is a commutator, so that should presumably imply that every element has that form for some $a$ ). For $\psi,$ we want to characterize matrices of the form $y = x (x^{-1})^t$ (by dimension counting this is a proper subset; it is pretty clear that it contains the complex orthogonal group).

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the automorphism group of $G$ is generated by inner automorphisms and $\psi(x) = x (x^{-1})^t.$ For inner $\phi,$ we are asking which $x$ satisfy the quadratic equation $x = [a, y]$ for a fixed a (if $a$ is allowed to vary, it is well-known that every element of a complex semi-simple lie group is a commutator, so that weakening is not interesting). For $\psi,$ we want to characterize matrices of the form $y = x (x^{-1})^t$ (by dimension counting this is a proper subset; it is pretty clear that it contains the complex orthogonal group).

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the automorphism group of $G$ is generated by inner automorphisms and $\psi(x) = x (x^{-1})^t.$ For inner $\phi,$ we are asking which $x$ satisfy the quadratic equation $x = y a y a^{-}]$ for a fixed a (if $a$ is allowed to vary, it is well-known that every element of a complex semi-simple lie group is a commutator, so that should presumably imply that every element has that form for some $a$ ). For $\psi,$ we want to characterize matrices of the form $y = x (x^{-1})^t$ (by dimension counting this is a proper subset; it is pretty clear that it contains the complex orthogonal group).