LaSalle's invariance principle is a commonly used extension of Lyapunov's second method to study asymptotic stability of nonlinear systems. If the system can be written in polynomial form, the examination can be automated using algebraic geometry and quantifier elimination. This article addresses this automated examination using a method relying on polynomial ideals and applies it on some example systems. In addition, some properties of these special ideals are derived that allow to reduce the computational effort significantly.