Let $k$ be a field of characteristic $p > 0$ (assume $k$ is perfect if it helps). Let $G$ be a connected finite group scheme of height one over $k$. Then $G$ is determined by its Lie algebra $\mathfrak{g}$, as a restricted Lie algebra: it can be recovered as the spectrum of the Hopf algebra dual to the restricted universal enveloping algebra $U( \mathfrak{g} )$.

Let $R$ be a local Artinian $k$-algebra with residue field $R / \mathfrak{m} \simeq k$. I would like to understand the category of $G$-torsors over $R$ (with respect to the flat topology, say) which are trivialized over $R / \mathfrak{m}$. This category feels very ``infinitesimal,'' so it seems reasonable to expect that there is a way to describe its objects concretely in terms of linear-algebraic data related to $\mathfrak{g}$ (and without ever mentioning the flat topology). Can this be done? (And if so, how?)