  We propose a systematic elementary approach based on trajectories to prove
Poincar\'e inequalities for hypoelliptic equations with an arbitrary number of
H\"ormander commutators, both in the local and in the non-local case. Our
method generalises and simplifies the method introduced in Guerand-Mouhot
(2022) in the local case with one commutator, and extended in Loher (2024) to
the non-local case with one commutator. It draws inspiration from the paper
Niebel-Zacher (2022), although we use different trajectories. We deduce Harnack
inequalities and H\"older regularity along the line of the De Giorgi method.
Our results recover those in Anceschi-Rebucci (2022) in the local case, and are
new in the non-local case.
