Robert Gilmore's Lie Groups, Lie Algebras and Some of Their Applications

has a physics projection of Pauli matrices to SL(2,C) used in a Nuclear magnetic resonance calculations.

He leaves out the 4th dimension,

so I thought of this Minkowski Determinant {1,1,1,-1} signature group as a 4d projection.

I get a very neat 4 cusp fractal in 3 group by using a disk matrix

{{1 + I, 1}, {1, 1 - I}}

as the third Kleinian group matrix:

s[1] = {{1 + I, 1}, {1, 1 - I}}; s[2] = {{0.5+ 0.5 I, 0.7071067811865475}, {-0.7071067811865475, 0.5- 0.5 I}}; s[3] = {{0.5- 0.5 I, -0.7071067811865475}, {0.7071067811865475, 0.5+ 0.5 I}}; s[4] = Inverse[s[1].s[2].s[3]];

(* Mathematica*) (Minkowski 4d 2x2 group)

s[1] = {{1, 0}, {0, 1}};

s[2] = {{I, 0}, {0, -I}};

s[3] = {{0, 1}, {-1, 0}};

s[4] = {{0, 1}, {1, 0}};

In[24]:= Clear[a, b, c, d]

m = as[1] + bs[2] + cs[3] + ds[4];

In[26]:= d = d /. Solve[Det[m] == 1, d][[2]]

Sqrt[-1 + a^2 + b^2 + c^2]

(Unit sphere projection)

a = Sin[t]*Sin[p]; b = Cos[t]*Sin[p]; c = Cos[p];

(* two matrices at{Pi/4,3Pi/4} 90 degress apart*)

N[m /. t -> Pi/4 /. p -> Pi/4] {{0.5 + 0.5 I, 0.707107}, {-0.707107, 0.5 - 0.5 I}}

N[m /. t -> 3Pi/4 /. p -> 3Pi/4] {{0.5 - 0.5 I, -0.707107}, {0.707107, 0.5 + 0.5 I}}

(end)