Conjugacy classes of Cartan subgroups for the simply connected split group of type F4 Cartan #0: split: 4; compact: 0; complex: 0 imaginary root system is empty real root system: F4 complex factor is empty twisted involution orbit size: 1152/1152 = 1 size of parameter space fiber: 1.16 = 16 Cartan #1: split: 2; compact: 0; complex: 1 imaginary root system: A1 real root system: C3 complex factor is empty twisted involution orbit size: 1152/96 = 12 size of parameter space fiber: 1.4 = 4 Cartan #2: split: 2; compact: 0; complex: 1 imaginary root system: A1 real root system: B3 complex factor is empty twisted involution orbit size: 1152/96 = 12 size of parameter space fiber: 1.4 = 4 Cartan #3: split: 1; compact: 1; complex: 1 imaginary root system: B2 real root system: B2 complex factor is empty twisted involution orbit size: 1152/64 = 18 size of parameter space fiber: 2.2 = 4 Cartan #4: split: 0; compact: 0; complex: 2 imaginary root system: A1.A1 real root system: A1.A1 complex factor is empty twisted involution orbit size: 1152/16 = 72 size of parameter space fiber: 1.1 = 1 Cartan #5: split: 0; compact: 2; complex: 1 imaginary root system: C3 real root system: A1 complex factor is empty twisted involution orbit size: 1152/96 = 12 size of parameter space fiber: 4.1 = 4 Cartan #6: split: 0; compact: 2; complex: 1 imaginary root system: B3 real root system: A1 complex factor is empty twisted involution orbit size: 1152/96 = 12 size of parameter space fiber: 4.1 = 4 Cartan #7: split: 0; compact: 4; complex: 0 imaginary root system: F4 real root system is empty complex factor is empty twisted involution orbit size: 1152/1152 = 1 size of parameter space fiber: 16.1 = 16 total number of parameters for this inner class: 16 + 48 + 48 + 72 + 72 + 48 + 48 + 16 = 368