The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [Meerbergen and Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1982- 1999] avoids this computation, instead performing an inverse iteration for a certain set of real eigenvalues and that requires the solution of a large-scale Lyapunov equation at each iteration. In this study, we refine the Lyapunov inverse iteration method to make it more robust and efficient, and we examine its performance on challenging test problems arising from fluid dynamics. Various implementation issues are discussed, including the use of inexact inner iterations and the impact of the choice of iterative solution for the Lyapunov equations, and the effect of eigenvalue distribution on performance. Numerical experiments demonstrate the robustness of the algorithm.