Because SU(2) we get a lot of interesting phenomena, including that there are two types of particles, bosons and fermions. We get some interesting phenomena that only rotating by 720deg (two full rotations) bring back to the initial state. And I am not talking only about USB-A, but about spinors (https://en.wikipedia.org/wiki/Spinor) - there are some party tricks around that (vide https://www.reddit.com/r/physicsmemes/comments/181oldw/a_ger...).
sl(2, R) ≅ so(2,1)
sl(2, C) ≅ so(3,1)
sl(2, H) ≅ so(5,1)
sl(2, O) ≅ so(9,1)
Dirac equation is the C case, the other cases have their uses.
I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.
And Lewis Carroll (Oxford (Math)), preferred Euclidean geometry over quaternions;
Quaternions:
q = a + bi + cj + dk
-1 = i^2 = j^2 = k^2
> In a quaternion, if you lose the scalar (a) — the "real" or "time" component — you are left with only the three imaginary components (i, j, k) rotating endlessly in a circle.
(An exercise for learning about Lorentzian mechanics, then undefined: Rotate a cube about a point other than its origin. Then, rotate the camera about the origin.)
4D Quaternions (a + bi + cj + dk) are more efficient for computers than 3D+t Euclideans. Quaternions do not have the Gimbal Lock problem that Euclidean vectors have. Quaternions interpolate more smoothly and efficiently, which is valuable for interpolating between keyframes in a physical simulation.
Why are rotations and a scalar a better fit?
Quaternions were published by William Rowan Hamilton (Trinity,) in 1843, in application to classical mechanics and Lagrangian mechanics.
Maxwell's original ~20 equations are also quaternionic; things are related with complex rotations in EM field theory too. Oliver Heaviside then "simplified" those quaternionic expressions into accessible vectors.
Is there Gimbal Lock in the Heaviside-Hertz vector field reinterpretation of Maxwell's quaternionic EM field theory? Maxwell's has U(1) gauge symmetry.
History of quaternions: https://en.wikipedia.org/wiki/History_of_quaternions
There's a 1-1 mapping between complex numbers and 2D rotation matrices that only do rotation and scaling. The benefit is that the complex number only has two coefficients, not four like the matrix. Multiplying these complex numbers is the same as multiplying the equivalent matrices. Quaternions are the same idea just in 3 dimensions (so with 3 imaginary units i j k, not just i, one per plane).