Normal,unipotent subgroup schemes of reductive groups

Let G be a reductive group over a field k of characteristic p. Let $k^{\mathrm{sep}}$ be a separable closure of k. If $p\ne 2$, there exists a linear representation of G that is faithful and semisimple; moreover, any unipotent, normal subgroup scheme of G is trivial. For $p=2$, these two properties hold if and only if $G_{k^{\mathrm{sep}}}$ has no direct factor that is isomorphic to ${\mathbf{SO}}_{2n+1}$ for some $n?1$. To cite this article: A. Vasiu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).