Surjectivity Criteria for p-adic Representations, Part I

We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.