This paper builds on earlier results which studied certain notions of observability and controllability for uncertain systems defined by integral quadratic constraints. These notions were motivated by realization questions for uncertain systems and relate to questions of whether a state is unobservable for all possible uncertainties and whether a state is controllable for some possible uncertainty. Originally, the notion of robust unobservability was characterized in terms of a linear quadratic optimal control problem but later a geometric characterization was obtained. Also, originally, the notion of possible controllability was characterized in terms of a linear quadratic optimal control problem, but in this paper it is also characterized geometrically and this is combined with the geometric characterization of robust unobservability to give a complete Kalman decomposition for uncertain linear systems.