an affine algebra $A$ over $K$, we define its radical index to be the smallest integer $n$, such that $f^n= 0$ for all $f \in$ Rad$({\bar A}) \overset{\text{by~defn}}{=}$ Rad $(A \otimes_{K} {\bar K})$. Now let $m \in M$ be a $K$-rational point of $M$ which is not semistable for the $G$-action.\
Recall that the the action of $G$ is said to be strongly seperable at a point $m \in M$ if the isotropy subgroup scheme at every ${\bar K}$-valued point in the closure of $O(m)$ is reduced, where $O(m)$ denotes the orbit of $m$. Let $P(m)$ be the instability parabolic of $m$. There exists $g \in G$ such that the parabolic $P = gP(m)g^{-1}$ is defined over $K_s$. By uniqueness of the instability parabolic and Galois descent, it is already defined over $K$. Let $x_m=gm$. Then the instability parabolic of $x_m$ is $P$. Since $P(x_m)$ is defined over $K$, it contains a maximal torus over $K$ (which is split over $K_s$). Hence there is a unique instability 1-PS of $x_m$ contained in this maximal torus which is defined over $K_s$ and hence by uniqueness defined over $K$.\
Consider the decomposition of $V = \oplus V_i$ into simultaneous eigenspaces for the 