Here's a similar argument from my naive, intuition-based perspective: It's important to remember that the the group of rotations in 3d space is represented by

Here's a similar argument from my naive, intuition-based perspective: It's important to remember that the the group of rotations in 3d space is represented by _unit_ quaternions. This is a subset of the full 4d quaternion space: a spherical shell around the origin, like the skin of a 3d ball but in 4d. A 3d ball has a 2d, "flat" skin that is "spherically" symmetric. A 4d ball has a 3d, "volumetric" skin that is, similarly, "spherically" symmetric. If we are stuck to that 3d skin, it makes perfect sense that it manages to represent rotations in 3d, which have 3 degrees of freedom and spherical symmetry. The space of rotations that is formed by this "skin" has two special points: the point where all the imaginary coordinates are zero and the real coordinate is one, and it's dual, the negative one, precisely because we are talking about _unit_ quaternions. (Similarly as there is only two "purely x" points in a unit circle: (1, 0) and (-1, 0)). These points represent the identity rotation, i.e. "don't rotate at all". This makes sense, if you think how complex numbers work: the effect of multiplying "one" is that it keeps everything as-is, whereas every other "unit" complex number has a rotating effect. Then as you think the three imaginary dimensions as degrees of freedoms you can start traveling into, from this neutral point, you get all kinds of rotations. The geometry of the "round", "skin-shaped" space ensures that the rotations wrap around the correct way, "spherically". Especially that after you have travelled "tau" (2 times pi) units, you are again in purely "real", "not rotated" state. And the spherical symmetry means that this works similarly in _any_ direction you can travel to. The only gotcha is that the unit quaternion space contains doubly the space of minimal 3d rotations, because of the negative number symmetry. There's some good arguments why this is especially beautiful and true, but they are a bit beyond me.