∀ {K : Type u_2} {L : Type u_1} [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]
  (q : Submodule K (Module.Dual K ↥H)),
  (∀ (i : ↥LieSubalgebra.root), q ∈ Module.End.invtSubmodule ↑((LieAlgebra.IsKilling.rootSystem H).reflection i)) →
    ∀ {x m : L},
      m ∈ (↑(⨆ α, LieAlgebra.IsKilling.sl2SubmoduleOfRoot ⋯)).carrier →
        ⁅x, m⁆ ∈ (↑(⨆ α, LieAlgebra.IsKilling.sl2SubmoduleOfRoot ⋯)).carrier