Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. Importantly (for me at least), $c_\gamma(t)$ decomposes into linear factors (over $F$) if and only if $\gamma$ lives in a maximal split torus, and the splitting field $E_\gamma$ of $c_\gamma$ is the minimal field extension $E$ such that $\gamma$ lives in an $E$-split torus.

Is there an appropriate generalization to other split groups? For instance, is there a function $f_\gamma$ (or set of functions) in $k[G]^G$ such that the statement `$\gamma$ lives in an $F$-split maximal torus' is equivalent to some statement about the algebraic properties of $f_\gamma$?

More generally, if $G$ is quasi-split over $F$, is there a function $f_\gamma$ in $k[G]^G$ whose properties determine if $\gamma$ lives in a Borel defined over $F$? (This is a generalization of the previous, since if $G$ is split then $\gamma$ lies in a maximal split torus if and only if it lives in a Borel).

Just for the record, my first thought was to let $f_\gamma$ be the characteristic polynomial of $Ad(\gamma)$ acting on the Lie algebra $\mathfrak{g}$. This fails even over $GL_2$, where $F$ is any field not having a square root of $-1$. Then $\gamma = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ has $f_\gamma(t) = (t-1)^2(t+1)^2$, but $\gamma$ cannot be diagonalized over $F$.