The stability regions of input-saturated linear systems are examined, with attention focused on state-feedback LQ control. The results describe hyper-ellipsoidal domains, which are subsets of the actual domain of attraction for the system. This is not new; the novelty is in the use of the low-and-high gain technique, to reduce conservatism in the results. In addition to the standard regulator, the tracking case is treated and some comments are made regarding other extensions.