Why does a rigid body have angular momentum in its body-fixed frame?

[karlo1961]

The angular momentum of a rigid body is often calculated into different coordinate systems. One is an inertial system and the other is a body system moving with the rigid body. Now, in the body system the rigid body is at rest and has no movement at all. How is it that it can still have a non-zero angular momentum the body system.

 

[HallsofIvy]

Typically, the "system moving with the rigid body" is a system in which the center of mass is always at (0,0,0), not one that is rotating with the body!