Left-invariant distributions diffeomorphic to flat distributions

For a stratified group G , we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of G . Vice versa, we show that all Lie groups with a left-invariant distribution that is locally diffeomorphic to the flat distribution of G belong to the class we constructed, if the Lie algebra of G has finite Tanaka prolongation.