... ball.1
If you want to be really concrete, imagine a spinning gyroscope fitting snugly in a box. Rotate the box.

... (frames).2
In other words, a element of determines a mapping of to . As it happens, the action is faithful, i.e., the mapping determines the element of .

... moment.3
A quick review: write for the equivalence class of . We will associate either a complex number or else with each class . If , then , where . All pairs of the form belong to the class . So we associate with if , and with . Mapping the complex plane plus to the Riemann sphere via the usual stereographic projection completes the trick. Some sample points to bear in mind: the north pole is ; the south pole is ; points on the equator have the form . (For the purist, the special treatment of rankles. It is not singled out on either the complex projective line or on the Riemann sphere. Later I will show how to set up the correspondence without this blemish.)

... simple4
In the infinite dimensional case, you have to use the spectral decomposition of instead of an orthonormal basis of eigenvectors; another reason why spin is simpler than position.

... proportional to . 5
Why wouldn't the electron simply snap into alignment with the magnetic field? Answer: the spinning electron would act like a gyroscope, and precess in response to the torque exerted by the field. Thus it would maintain its angle of inclination to the field.