action of $\lambda$, where $V_i=\{v \in V \mid \lambda(t)(v)=t^i(v)\}$. Let $j=m(x_m,\lambda)$ and $V^j=\oplus V_i, i\geq j$. Define the $K_s$-scheme $M(P)_{x_m}$ to be the scheme theoretic intersection of the $K_s$-subscheme ${\mathbb P}(V^j)$ and $O(m)$ of ${\mathbb P}(V)$. The following proposition summarizes the basic properties of the scheme $M(P)_{x_m}$.

[\[strongly seperable\]]{} The $\bar K$-valued points of $M(P)_{x_m}$ are precisely those points in the $K$-scheme $O(m)$ for which the instability parabolic is $P(x_m)$. Also, when the action of $G$ on $m$ is strongly seperable, then $M(P)_{x_m}$ is absolutely reduced.

Suppose one can find a $K_s$ rational point $m'$ in $M(P)_{x_m}$, then by proposition [ \[strongly seperable\]]{}, its instability parabolic being $P(x_m)$ is hence defined over $K_s$. Since $m$ and $m'$ are both $K_s$ rational points, they are translates of each other by a $G(K_s)$-valued point $g$ and hence their instability parabolic are conjugates by $g$. This will prove that the instability parabolic for $m$ is defined over $K_s$ and hence by uniqueness and Galois descent it is defined over $K$.\
Thus the problem of showing the existence of the integer $t$ in proposition [\[field of definition\]]{} boils down to finding a finite purely inseperable extension $L$ of $K_s$ (independent of the point $m$) 