stuck on derivation of Euler's equations in rigid body dynamics

bigerst

i was reading about derivation of Euler's equations for rotational dynamics (john taylor, classical mechanics, chapter 10) when i got stuck on one of the reasonings
essentially it refers to the moment of inertia tensor, since the tensor itself about a point is dependent on the position of the rigid body (hence the tensor would have to be a function of time) hence it is argued that it would be helpful to construct a "body frame" fixed within the body with its axis pointing at the principal axis of rotation of the body, this body frame would rotate along with the body (since it is fixed within the body). then came the part i didnt understand, it said the angular momentum measured in the body frame is L = [I] w, where [I] denotes the diagonal matrix and w the angular velocity. however since the frame is fixed inside the body, should the rigid body be stationary relative to the frame and hence w is zero? what is wrong with my reasoning?