The rotation group

Hold up your hand. Turn it once.

That motion — that single rotation — is an element of a group called SO(3).

The special orthogonal group in three dimensions.

Every rotation of every object in space is an element of SO(3). Not a postulate. The definition of rotation in three dimensions.

Three dimensions is not a postulate either. Notebook II Chapter 3 derived N = 3 from the four independent axioms.

The rotation group is what those three dimensions force.