Let $g$ be an element of a linear algebraic group with Jordan decomposition $g=su=us$. Is there a general way of relating the set of conjugates of $g$ to the conjugates of $s$ and $u$? (denoted by $g^G$)?
The set $\{u's' : u'\in u^G, s'\in s^G\}$ seems to be larger than $x^G$. But maybe something like $$x^G = \{u's' : u'\in u^G, s'\in s^G, u's'=s'u'\}?$$
(The inclusion $\subseteq$ is clear, what about $\supseteq$?)