How Do You Derive Euler's Equation of Motion for a Rigid Body?

[D H]

Does this mean that relative to any other frame, $T' = \frac{1}{2}(I_1'w_1'^2 + I_2'w_2'^2 + I_3'w_3'^2)$

No. The generic formula is $T = \frac 1 2 \vec L \cdot \vec \omega = \frac 1 2 (I\vec \omega)\cdot \vec \omega$. This reduces to your simpler form only in the special case where the inertia tensor is diagonal.

One grad student I was talking to said that the topic of rigid body motion was overall more difficult to comprehend than some of the research he was working on. I guess that is not very encouraging, but it puts your point into perspective.

One reason it's confusing is because rotations in three dimensional space don't commute. I didn't attain a true understanding of rigid body motion until I learned about Lie groups.