Examples

For a uniform sphere, the cross-terms disappear and the moments of inertia are all the same, leaving $\tau_i=I\omega_i$ , for $i=1,2,3$ . Since any three orthogonal vectors can serve as eigenvectors of the moment of inertia tensor, we have that, for a uniform sphere, any three orthogonal axes can be chosen as principal axes.

For a cylinder that is not spinning about its axis, we similarly obtain two uncoupled equations $\tau_i=I\omega_i$ , for $i=1,2$ , given $\omega_3=\tau_3=0$ (no spin). Note, however, that if we replace the circular cross-section of the cylinder by an ellipse, then $I_1\ne I_2$ and there is a coupling term that drives $\dot{\omega}_3$ (unless $\tau_3$ happens to cancel it).