The Sub-Laplacian and the Twisted Laplacians
By Theorem 17.4 in the preceding chapter, the vector fields X and Y on the Heisenberg group H1 enjoy the special property that
[X,Y ] = −T. As such, by Theorem 17.3, the vector fields X and Y and their first-order commutators span the Lie algebra h1. So, the vector fields X and Y play a special role in the study of the Heisenberg group. They are known as the horizontal vector fields on H1, and T is referred to as the missing direction.