Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.

The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the discriminant of the characteristic polynomial is $\Delta=(b-a)^{2}$.

In the theory of algebraic groups we also have another discriminant to check if it's regular semisimple: $\Delta'(t)=\prod\limits_{\alpha\in R}(\alpha(t)-1)=\frac{(b-a)^{2}}{ab}$.

For instance, we can obtain it as the valuation of the determinant of $Id-Ad(t)$ on $\mathfrak{gl}_{2}(F)/\mathfrak{t}(F)$, and so $val (\Delta(t))\neq val (\Delta'(t))$.

The reason is that somehow the second definition kills the center and is the right one only for semisimple.

My question is: do we have a group theoretic interpretation of $\Delta$ which gives us the right determinant?

Of course, I stated it for $GL_{2}$ but the same question holds more generally for reductive groups.