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Something is symmetric if it is invariant under transformation. If I have a rotating disc, it is said to have rotational symmetry because the conserved quantity in such a system is angular momentum.

Is it proper to say that the Law of Conservation of Angular Momentum should hold regardless of an observer's frame of reference?

Or is it simply the fact that the physics of the rotational system are unchanged no matter what axis or angle you rotate something at?

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I'm not sure of the precise definition, what I can say is that, at least in the Standard Model case, is said that the symmetry is broken(the Higgs mechanism for example is a process of this kind) when you change the reference frame such that is not anymore explicit the conserved quantity, and thus the symmetry.

But although it's not explicit the angular momentum is still a conserved quantity, no matter the reference frame you are using, if it's not clear you'll see when calculating explicitly that is needed to take into account the transformation (translation, rotation...) done to change reference frames. I hope this clears something up :)

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This answer is for classical mechanics. Angular momentum is conserved only if there is no net external torque.

In a non-inertial (accelerating) frame of reference, fictitious forces and torques are necessary to describe the motion of a particle or a system of particles except for the special case where the center of mass (CM) is taken as the reference point.

If angular momentum is measured with respect CM of an object, the change in angular momentum is due to only the torques from real forces in the inertial frame even if the CM is accelerating, but this is a special case. Basic physics text assert this usually without proof (for example one of the physics texts by Halliday and Resnick). More advanced physics text, such as Mechanics by Symon, prove this starting from considering the change in angular momentum for a system of particles.

If you take a point other than the CM that is accelerating in a system of particles, to evaluate the change in angular momentum you must consider the torques due to the ficticious forces.

Physics mechanics text such as Mechanics by Symon or Classical Mechanics by Goldstein develop the laws of motion in detail in a non-inertial frame of reference.

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