(Ed Pegg suggested I publish this post here)
Exceptional group E8 tensor reduction on disk:
Example:Cartan matrix for exceptional group E8 (in cosmology the Universe model of an Dodecahedron is sometimes used)

e8 = {{2, -1, 0, 0, 0, 0, 0, 0},
  {-1, 2, -1, 0, 0, 0, 0, 0},
  {0, -1, 2, -1, 0, 0, 0, -1},
  {0, 0, -1, 2, -1, 0, 0, 0},
  {0, 0, 0, -1, 2, -1, 0, 0},
  {0, 0, 0, 0, -1, 2, -1, 0},
  {0, 0, 0, 0, 0, -1, 2, 0},
  {0, 0, -1, 0, 0, 0, 0, 2}}

The 2x8 octagon complex reduction tensor:

d={{1/2, I/2}, {0, I/Sqrt[2]}, {-(1/2), I/2}, {-(1/Sqrt[2]),
  0}, {-(1/2), -(I/2)}, {0, -(I/Sqrt[2])}, {1/2, -(I/2)}, {1/Sqrt[2],
  0}}

which gives:

m2x2= N[dt.e8.d]/Sqrt[Det[N[dt.e8.d]]]

 {{0. - 1.70434 I, -0.182993 + 0. I}, {-0.182993 + 0. I,
  0. + 0.606385 I}}

The scaled group is:

mu = N[1.0]
s[1] = {{mu + I, mu}, {mu, mu - I}}
s[2] = {{0.` - 1.7043411981641243` I, -0.1829927572190933` +
    0.` I}, {-0.1829927572190933` + 0.` I,
   0.` + 0.6063846548495646` I}}
s[3] = Inverse[s[1]]
s[4] = Inverse[s[2]]

Published notebook:
https://www.wolframcloud.com/obj/rlbagulatftn/Published/Nylander_%202group_E%20_%208_reduction%20_on%20_scaled%20_disk%20_limitset%20_%2011_%20%202d_sphere%20_on%20_Inverse%20_sphere.nb