$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic centralizer (not necessarily smooth) and $O_x$ for the orbit. Orbit-stabilizer holds, and implies $T_x(O_x) = \Lie(G/G_x)$.
Examples show that $\Lie(G_x)$ need not act trivially on $x$, but suggest that $\Lie(G/G_x) \cong \Lie(G)/\Lie((G_x)_{red})$: a tangent vector for $G_x$ will act trivially on $x$ if and only if it lies in the underlying reduced. This also matches classical intuition, where $(G_x)_{red}$ was taken as the stabilizer.
Is the isomorphism claimed above true under these hypotheses? Is there anything one can say over nonperfect fields, where $(G_x)_{red}$ may fail to be a subgroup of $G_x$ or $G$ at all?
