-- Module: Mathlib.Algebra.Lie.Weights.IsSimple
Goal:
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
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α
h_pairing_zero : S.pairing i j = 0
h_pos_zero : ⁅x_χ, m_pos⁆ = 0
h_neg_zero : ⁅x_χ, m_neg⁆ = 0
⊢ ⁅x_χ, (↑y : L)⁆ = 0

Tactic:
have h_bracket_zero : ⁅x_χ, (y : L)⁆ = 0 := by
    have h_chi_coroot_zero : χ (coroot α) = 0 := by
      have h_pairing_eq : S.pairing i j = i.1 (coroot j.1) := by
        rw [rootSystem_pairing_apply]
      rw [h_pairing_zero] at h_pairing_eq
      have w_eq {w₁ w₂ : Weight K H L} (h : w₁.toLinear = w₂.toLinear) : w₁ = w₂ := by
        apply Weight.ext; intro x; exact LinearMap.ext_iff.mp h x
      have hi_val : i.1 = χ := w_eq (by rw [← hi]; rfl)
      have hj_val : j.1 = α := w_eq (by rw [← hj]; rfl)
      rw [hi_val, hj_val] at h_pairing_eq
      exact h_pairing_eq.symm
    have h_lie_eq_smul : ⁅(y : L), x_χ⁆ = χ y • x_χ := lie_eq_smul_of_mem_rootSpace hx_χ y
    have h_chi_h_zero : χ y = 0 := by
      obtain ⟨c, hc⟩ := Submodule.mem_span_singleton.mp <| by
        rw [← coe_corootSpace_eq_span_singleton α, LieSubmodule.mem_toSubmodule]
        exact hy
      rw [← hc, map_smul, h_chi_coroot_zero, smul_zero]
    have h_bracket_elem : ⁅x_χ, (y : L)⁆ = 0 := by
      rw [← lie_skew, h_lie_eq_smul, h_chi_h_zero, zero_smul, neg_zero]
    exact h_bracket_elem