 Spectral Hopf condition for a cubic characteristic polynomial

A real cubic $p(\lambda )={\lambda }^3+A{\lambda }^2+B\lambda +C$ has roots $-A,\pm i\sqrt{B}$ when $B>0$ and $AB=C$. For $A>0$ this gives a stable transverse third direction. To conclude a generic Hopf bifurcation as a parameter varies, also require a transverse crossing of the imaginary pair and a nonzero first Lyapunov coefficient. The algebraic imaginary-root condition alone is not sufficient.