I believe the answer to your question is yes, at least if $G$ is a simple, simply-connected algebraic group, and if the characteristic is good for $G$ (and does not divide $n+1$ in type $A_n$). Check out the paper by Friedlander and Parshall, Rational actions associated to the adjoint representationRational actions associated to the adjoint representation, in Ann. scient. Ec. Norm. Sup., 4e serie, tome 20, no 2 (1987), p. 215-226.
LSpice
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