∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
  [inst_4 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_5 : H.IsCartanSubalgebra]
  [inst_6 : LieAlgebra.IsKilling K L] [inst_7 : LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsSimple K L],
  (LieAlgebra.IsKilling.rootSystem H).IsIrreducible