The horizontal distribution in the **Heisenberg group** is the kernel of the standard contact form:
$$
\alpha = dt + 2 \sum\_{j=1}^n (x\_j \, dy\_j - y\_j \, dx\_j).
$$



> 
> **Question.** Can one describe horizontal distribution in any **Carnot group** in terms of kernels of some $1$-forms?
> 
> 
> 


I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.



Heisenberg groups, Carnot groups and contact forms
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.


On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.


