Bases for Lie algebras and a continuous CBH formula

It is natural to search for solution formulas for x(t) = x(t, u) that, separate the time-dependent contributions of the controls u from the invariant, geometric role of the vector fields fk. Ideally, one may be able to a-priori obtain closed-form expressions for the flows of certain vector fields. The quadratures of the control might be done in real-time, or the integrals of the controls may be considered new variables for theoretical purposes such as path-planning or tracking. To make this scheme work, one needs simple formulas for assembling these pieces to obtain the solution curve x(t, u). Such formulas are nontrivial since in general the vector fields fk do not commute: already in the case of linear