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I have been told this is true for the moon on this question:

Angular momentum of orbiting + rotating body

but I don't understand why it should work. Surely the axis must be chosen such that the entire body is rotating around it? Otherwise you could come up with any answer?

EDIT: I think the we must define angular momentum about a point, such as the centre of orbit, and not an axis, such as the axis of orbit. So for a body rotating about an axis passing through this point: $L = w_i \times{r_i}\times{w_i}$ over all $i$ but $w_i$ is constant, so $L = wr_i.r_i - r^2\cdot{w}$ which is $Iw$ where $I$ is the moment of inertia about axis of rotation.

we have to show that this works for when the axis is shifted away from the origin so that $w_i$ is no longer constant but undergoes a transformation depending on $r_i$, if this fact is indeed true.

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