  We provide a study of the Riesz transforms on stratified nilpotent Lie
groups, and obtain a certain version of the pointwise lower bound of the Riesz
transform kernel. Then we establish the characterisation of the BMO space on
stratified nilpotent Lie groups via the boundedness of the commutator of the
Riesz transforms and the BMO function. This extends the well-known Coifman,
Rochberg, Weiss theorem from Euclidean space to the setting of stratified
nilpotent Lie groups. In particular, these results apply to the well-known
example of the Heisenberg group. As an application, we also study the curl
operator on the Heisenberg group and stratified nilpotent Lie groups, and
establish the div-curl lemma with respect to the Hardy space on stratified
nilpotent Lie groups.
