 Body-frame and space-frame frequencies of free symmetric-top precession

For principal inertias $I_{\perp },I_{\perp },I_{\parallel }$, the transverse angular-velocity components rotate in body axes at signed rate $(I_{\parallel }-I_{\perp }){\omega }_3/I_{\perp }$. The symmetry axis precesses about the inertially fixed angular momentum at rate $L/I_{\perp }$. These are different frame-dependent descriptions of the same motion.