On exponentiation and infinitesimal one-parameter subgroups of reductive groups

Let G be a reductive algebraic group over an algebraically closed field k of characteristic p>0, and assume p is good for G. Let P be a parabolic subgroup with unipotent radical U. For r⩾1, denote by Ga(r) the r-th Frobenius kernel of Ga. We prove that if the nilpotence class of U is less than p, then any embedding of Ga(r) in U lies inside a one-parameter subgroup of U, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if p is at least as big as the Coxeter number of G, then the cohomological variety of G(r) is homeomorphic to the variety of r-tuples of commuting elements in N1(g), the [p]-nilpotent cone of Lie(G).