Apparently this construction always provides the (outer) semi-direct product of the Lie group by its Lie algebra with respect to the group adjoint. This also m

Apparently this construction always provides the (outer) semi-direct product of the Lie group by its Lie algebra with respect to the group adjoint. This also means we should get another one considering body-fixed velocities instead, which I suspect will be some form of outer semi-direct product with respect to the coadjoint representation of the group.