Solution to commutator equation in semisimple algebraic group $\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and $\SL_n$ and $\GL_n$ the special and general linear groups over $K$. Let $\Phi \in \GL_n(K), H \in \SL_n(K)$. Is there always a solution $X \in \SL_n(K)$ to the equation $X\Phi X^{-1}\Phi^{-1} = H$?
If needed, I am happy to assume that $\Phi$ is semisimple and to only ask for a solution in a finite extension of $K$. If you have a reference for general semisimple groups, that would be wonderful.
I believe this goes by the general name of word equations in groups with constants, and I am aware of the theorem that for semisimple $G$ the commutator is the whole of $H$, but I can't seem to find the precise answer to my question in any of the references I've seen.
 A: $\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$EDIT: The below answers a slightly different question of whether a matrix in $\SL(n,K)$ is a commutator. The question by OP asks whether for fixed $\Phi$, every matrix in $\SL(n,K)$ is a commutator $\Phi A \Phi^{-1} A^{-1}$. The comment by YCor shows that the answer is negative.

Let $K$ be a field.
The paper "Thompson, R. C., Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101 (1961), 16–33." (link) claims the following theorems, quote:

Theorem 1. Let $\rho I_n \in \SL(n, K)$. Then $\rho I_n$ is always a commutator of
$\GL(n, K)$. Moreover, $\rho I_n$ is a commutator of $\SL(n, K)$ unless $\rho$ is a primitive $n$th root of unity in $K$ and $n \equiv 2 \mod{4}$.
In this exceptional case $\rho I_n$ can always be expressed as a product of two commutators of $\SL(n, K)$ and can be expressed as a single commutator of $\SL(n, K)$ when, and only when, the equation $-1 =x^2+y^2$ possesses a solution $x,y \in K$. This condition is always satisfied when $K$ has characteristic different from zero.


Theorem 2. Let $A \in \SL(n, K)$. If $A$ is not scalar and if $K$ has at least four elements, then $A$ is a commutator of $\SL(n, K)$.

