The group U(1)xSU(2) can be represented using all four parts of a quaternion, three in the exponential, the fourth as a normalized quaternion. The group covers the entire unit sphere, but has a bias for the past.
Quaternions do not commute in general, but they will commute if two quaternions point in the same direction. The common way to represent the group U(1) is with a normalized complex number. The same thing can be done with a quaternion. This will commute with a unitary quaternion if they both use the same quaternion pointing in the same direction. Electroweak symmetry uses all the degrees freedom available in a quaternion.
The group of unitary quaternions is the symmetry underlying the weak force of radioactive decay. Who would have thought the symmetry looks like this?
If one takes the vector part of a quaternion and takes the exponential, the norm is always equal to 1. The animation starts out at 8 points, those for exp(0, +/-1, +/-1, +/-1). These points grow into each other until they form a sphere. That sphere then shrinks to the point (1, 0, 0, 0), the furthest into the future the exponential can reach.
