Conjecture I for unirational algebraic groups over imperfect fields

Using the recent advances in the structure of algebraic groups over imperfect fields, we prove a generalization of Serre’s Conjecture I and of results that revolve around it. In particular, we show that the first Galois cohomology set of any unirational algebraic group is trivial if the cohomological dimension of the field is at most 1 in Kato’s sense.