We revisit the problem of semiglobal stabilization of linear discrete-time systems subject to input saturation and give an algebraic Riccati equation (ARE)-based approach to the proof of a fact we established earlier [3], i.e., a linear discrete-time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open-loop poles are located inside or on the unit circle. Moreover, we drastically relax the requirements on the characteristic of the saturation elements as imposed in our earlier work [3].