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The main idea is to avoid indices ( index and matrix notation for tensors)and avoid picking a nontatural frame or a specific basis. Instead differential geometry , lie groups, geometric algebra etc. should be used .

what is a nontatural frame?

You can find a Lie algebra/group treatment of the rigid body equations in

Bloch, A.M., Crouch, P.E., Marsden, J.E. and Ratiu, T.S., The symmetric representation of the rigid body equations and their discretization, Nonlinearity, 15 (2002), 1309.

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