For the kinematics of rotation in three dimensions, see quaternions and spatial rotation.

By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation.

However, quaternions have had a revival since the late 20th Century, primarily due to their utility in describing spatial rotations.

For further elaboration on modeling three-dimensional vectors using quaternions, see quaternions and spatial rotation.

This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts.

In representation theory, this corresponds to decomposing perturbations under the group of spatial rotations.

For ferromagnetic materials, the underlying laws are invariant under spatial rotations.

That is to say, any spatial rotation can be decomposed into a combination of principal rotations.

(For a description of this homomorphism see the article on quaternions and spatial rotations.)

Other types of spatial rotations are described in the article Rotation symmetry.