This technical note addresses the stabilization of switching discrete-time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.