Consider a non-planar rigid body rotating about a fixed axis (say, the Z-axis, chosen vertically). Let the origin $O$ is chosen somewhere on the Z-axis. Let $\textbf{r}_i$ represent the position vector of the $i^{th}$ particle of the rigid body. Then, by definition, the angular momentum of the body about $O$ is given by $$\textbf{L}=\sum\limits_{i}m_i\textbf{r}_i\times(\vec{\omega}\times\textbf{r}_i)=\sum\limits_{i}m_i[r_i^2\vec{\omega}-(\textbf{r}_i\cdot\vec{\omega})\textbf{r}_i].$$ Since $\vec{\omega}=\omega \hat{k},$ $$\textbf{L}=\sum\limits_{i}m_i\omega[-(z_ix_i)\hat{i}+(-z_iy_i)\hat{j}+(x_i^2+y_i^2)\hat{k}].$$ The Z-component of the angular momentum is $L_z=\sum\limits_{i}m_id_i^2\omega=I\omega$ is usually treated with a special importance, (also to $\tau_z=\frac{dL_z}{dt}=I\dot{\omega}$).
Why are the dynamics of the other components, $L_x$ and $L_y$, not considered (in school-level textbooks such as Halliday, Resnick and Walker) even though they are nonzero, and may change if a force $\textbf{F}$ in an arbitrary direction is applied (because the torque $\vec{\tau}=\textbf{r}\times\textbf{F}$ will, in general, have all components nonzero)?
