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• Previous prediction of the steady-oscillatory transition in published works has been summarized, and we find the transition point has yet to come to consensus. • The critical Reynolds number is predicted numerically by a present method SCM-ACM with high accuracy and fewer nodes. • The temporal evolution of velocity, the amplitude evolution, and the distribution of dimensionless angular frequency are studied. Massive computations of three-dimensional (3D) flow in subcritical flow state are carried out to investigate the transition from 3D steady flow to time-periodic oscillatory flow via Hopf bifurcation in a cube. Previous prediction of the steady-oscillatory transition in published works has been summarized. The transition point in lid-driven cavity has yet to come to consensus as the critical parameters are predicted in a wide range of values. The 3D unsteady governing equations are solved numerically using the spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), SCM-ACM, a combined method with high accuracy that we developed. The temporal velocity evolution, amplitude evolution and dimensionless angular frequency are investigated. The accurate critical Reynolds number R e c r is predicted by Richardson extrapolation. Results show that the initial conditions have significant effect to obtain the velocity which gradually approaches to steady state in the subcritical flow state, and appropriate initial conditions are given to efficiently obtain stable results by gradually increasing R e. Exponential decay of velocity amplitude becomes weaker with the increase of R e , and the decay rate is strictly linear against R e , while the dimensionless angular frequency remains constant. The R e c r is affected by the number of nodes, which has also been observed by Feldman and Gelfgat [1] and Kuhlmann and Albensoeder [2] , while the critical frequency is unaffected by the number of nodes in the present results. It also indicates that the R e c r and the constant dimensionless angular frequency can be innovatively predicted with more accuracy and much fewer nodes. The improved point of steady-oscillatory transition in 3D lid-driven cube is R e c r = 1916.6 via Hopf bifurcation by Richardson extrapolation with dimensionless angular frequency ω = 0.5752. [ABSTRACT FROM AUTHOR]