On Controllability of Driftless Control System on Symmetric Spaces

The paper states an “Extended Rashevsky–Chow” theorem on symmetric spaces: if the Lie triple system generated by the control fields spans TxS everywhere, then the driftless system is globally controllable. It sketches a proof à la Jean by constructing compositions of flows to realize double Lie brackets [[A,B],C], showing local accessibility and then global controllability on a connected S. The model instead observes that the Lie–triple closure is contained in the usual Lie algebra generated by the vector fields, so the hypothesis implies the standard Lie Algebra Rank Condition; Rashevsky–Chow (or Sussmann’s orbit theorem) then yields global controllability, and a time-rescaling argument gives reachability in any prescribed tf>0 with L2 controls. Both arguments are valid; the model’s proof is simpler and uses standard tools, while the paper’s proof follows a triple-bracket flow construction.