Strong classification of the quartic homogeneous layer

Construct a basis of the free Lie algebra generated by X_0 and X_1 that strongly classifies the homogeneous component consisting of brackets with four occurrences of X_1, by making smooth small-time local controllability sufficient when the good brackets span the state space and necessary compensation involve only good brackets.

Background

The basis B established in the paper weakly classifies the quartic layer S_{1,4}(X), but it does not strongly classify that layer because competitions between bad quartic brackets can nevertheless produce smooth-STLC. Strong classification would require a partition into good and bad brackets such that vanishing of the bad brackets is sufficient for controllability and every bad bracket is compensated solely by good brackets.

The problem addresses the resulting nonuniqueness of weak classifications and seeks a basis-independent separation of controllable and obstructing directions. The authors note that a solution might require bases more general than Hall bases, despite the favorable algebraic and control-theoretic properties of Hall bases.