suppose we need to calculate $\overrightarrow{L}$ about an axis, but the rigid body is not rotating about this axis. Can we define the $\overrightarrow{L}_{axis}$ still? I think we should be able to since $\overrightarrow{L}$ (for an infinitesimal point on the rigid body) is just $dm(\overrightarrow{r}\times\overrightarrow{v})$
If it is possible then is there a way to determine angular momentum in such cases more systematic approach (with proof) for
Special case where we have to only deal with axis parallel to axis of rotation
The general case i.e. the axis of rotation is tilted/skewed w.r.t to axis of rotation.