Issues of asymptotic stabilization of a class of non-linear driftless systems are presented. In addition to the necessary and sufficient condition for the existence of a smooth time-invariant asymptotic stabilizer, sufficient condition for the existence of a quadratic-type Lyapunov function candidate is also proposed herein to alleviate the construction of stabilizing control laws. Following the deduction of the equivalence of the sufficient condition and the determination of the local definiteness of a defined scalar function, the stabilizability checking conditions are then derived in terms of system dynamics and its derivatives at the origin only. These are achieved by taking Taylor's series expansion on system dynamics. The derived conditions are shown to be consistent with those obtained by Brockett. Comparative results of Liaw and Liang are also included. Finally, examples are given to demonstrate the use of the main results.