Lifting Lie algebra on certain decoupled boundaries

Inspired by the celebrated work of Rothschild and Stein, we investigate explicit liftings of some smooth vector fields satisfying the bracket-generating condition with step depending on the point, to Lie algebras. The examples we focus on are certain decoupled domains introduced by Nagel and Stein, whose boundaries provide a natural setting for such constructions. We present explicit models for the lifted Lie algebras associated with these boundaries and analyze their structural properties.