The classical definition for angular momentum is L = r x p, which works well for a particle or any object that is internally at rest, or at least not rotating but it fails if the particle is rotating. For example, the Earth’s angular momentum around the Sun should not only include the orbital angular momentum but also Earth's rotation angular momentum.
For a complex system we can define the average momentum as: $$\boldsymbol{p_{avg}}=\int {\textbf{p}(\textbf{x})dx}=\int {\rho(\textbf{x})\textbf{v}(\textbf{x})dx}$$
This way we can define the external average momentum as: $$\boldsymbol{L_{ext}}=\textbf{r}\times \boldsymbol{p_{avg}}$$
We can define the rest inertial reference system as the one where the particle has $\boldsymbol{p_{avg}}$, then from this rest system, we can define the internal angular momentum for the object as: $$\boldsymbol{L_{int}}=\int {\mathbf{r(x)}\times \mathbf{p(x)}d\mathbf{x}}$$
My question is, can this internal angular momentum be added when measuring the angular momentum from another reference system? That is, can we express $\boldsymbol{L_{Earth}}=\boldsymbol{L_{Orbit}}+\boldsymbol{L_{Rotation}}$, having calculated the Rotation from the Earth’s rest system and the Orbit from the Sun’s rest system, or should we do some extra operation?
I have been looking for sources to confirm this but I have not found any.
