Conjugacy classes of Cartan subgroups for the simply connected split group of type E8 Cartan #0: split: 8; compact: 0; complex: 0 imaginary root system is empty real root system: E8 complex factor is empty twisted involution orbit size: 696729600/696729600 = 1 size of parameter space fiber: 1.256 = 256 Cartan #1: split: 6; compact: 0; complex: 1 imaginary root system: A1 real root system: E7 complex factor is empty twisted involution orbit size: 696729600/5806080 = 120 size of parameter space fiber: 1.64 = 64 Cartan #2: split: 4; compact: 0; complex: 2 imaginary root system: A1.A1 real root system: D6 complex factor: A1 twisted involution orbit size: 696729600/184320 = 3780 size of parameter space fiber: 1.16 = 16 Cartan #3: split: 2; compact: 0; complex: 3 imaginary root system: A1.A1.A1 real root system: A1.D4 complex factor: A2 twisted involution orbit size: 696729600/18432 = 37800 size of parameter space fiber: 1.4 = 4 Cartan #4: split: 0; compact: 0; complex: 4 imaginary root system: A1.A1.A1.A1 real root system: A1.A1.A1.A1 complex factor: A3 twisted involution orbit size: 696729600/6144 = 113400 size of parameter space fiber: 1.1 = 1 Cartan #5: split: 2; compact: 2; complex: 2 imaginary root system: D4 real root system: D4 complex factor: A2 twisted involution orbit size: 696729600/221184 = 3150 size of parameter space fiber: 4.4 = 16 Cartan #6: split: 0; compact: 2; complex: 3 imaginary root system: D4.A1 real root system: A1.A1.A1 complex factor: A2 twisted involution orbit size: 696729600/18432 = 37800 size of parameter space fiber: 4.1 = 4 Cartan #7: split: 0; compact: 4; complex: 2 imaginary root system: D6 real root system: A1.A1 complex factor: A1 twisted involution orbit size: 696729600/184320 = 3780 size of parameter space fiber: 16.1 = 16 Cartan #8: split: 0; compact: 6; complex: 1 imaginary root system: E7 real root system: A1 complex factor is empty twisted involution orbit size: 696729600/5806080 = 120 size of parameter space fiber: 64.1 = 64 Cartan #9: split: 0; compact: 8; complex: 0 imaginary root system: E8 real root system is empty complex factor is empty twisted involution orbit size: 696729600/696729600 = 1 size of parameter space fiber: 256.1 = 256 total number of parameters for this inner class: 256 + 7680 + 60480 + 151200 + 113400 + 50400 + 151200 + 60480 + 7680 + 256 = 603032