 Because formula_89 is an arbitrary vector, from nondegeneracy of scalar product follows

Since the spin angular velocity tensor of a rigid body (in its rest frame) is a linear transformation that maps positions to velocities (within the rigid body), it can be regarded as a constant vector field.In particular, the spin angular velocity is a Killing vector field belonging to an element of the Lie algebra SO(3) of the 3-dimensional rotation group SO(3).Also, it can be shown that the spin angular velocity vector field is exactly half of the curl of the linear velocity vector field v(r) of the rigid body.