/-- See also `bourbaki1975b` Chapter VII §1.1, Proposition 2 (ii). -/
protected theorem weight_vector_multiplication (M₁ M₂ M₃ : Type*)
    [AddCommGroup M₁] [Module R M₁] [LieRingModule L M₁] [LieModule R L M₁] [AddCommGroup M₂]
    [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂] [AddCommGroup M₃] [Module R M₃]
    [LieRingModule L M₃] [LieModule R L M₃] (g : M₁ ⊗[R] M₂ →ₗ⁅R,L⁆ M₃) (χ₁ χ₂ : R) (x : L) :
    LinearMap.range ((g : M₁ ⊗[R] M₂ →ₗ[R] M₃).comp (mapIncl 𝕎(M₁, χ₁, x) 𝕎(M₂, χ₂, x))) ≤
      𝕎(M₃, χ₁ + χ₂, x) := by
  -- Unpack the statement of the goal.
  intro m₃
  simp only [TensorProduct.mapIncl, LinearMap.mem_range, LinearMap.coe_comp,
    LieModuleHom.coe_toLinearMap, Function.comp_apply, Pi.add_apply, exists_imp,
    Module.End.mem_maximalGeneralizedEigenspace]
  rintro t rfl
  -- Set up some notation.
  let F : Module.End R M₃ := toEndomorphism R L M₃ x - (χ₁ + χ₂) • ↑1
  -- The goal is linear in `t` so use induction to reduce to the case that `t` is a pure tensor.
  refine t.induction_on ?_ ?_ ?_
  · use 0; simp only [LinearMap.map_zero, LieModuleHom.map_zero]
  swap
  · rintro t₁ t₂ ⟨k₁, hk₁⟩ ⟨k₂, hk₂⟩; use max k₁ k₂
    simp only [LieModuleHom.map_add, LinearMap.map_add,
      LinearMap.pow_map_zero_of_le (le_max_left k₁ k₂) hk₁,
      LinearMap.pow_map_zero_of_le (le_max_right k₁ k₂) hk₂, add_zero]
  -- Now the main argument: pure tensors.
  rintro ⟨m₁, hm₁⟩ ⟨m₂, hm₂⟩
  change ∃ k, (F ^ k) ((g : M₁ ⊗[R] M₂ →ₗ[R] M₃) (m₁ ⊗ₜ m₂)) = (0 : M₃)
  -- Eliminate `g` from the picture.
  let f₁ : Module.End R (M₁ ⊗[R] M₂) := (toEndomorphism R L M₁ x - χ₁ • ↑1).rTensor M₂
  let f₂ : Module.End R (M₁ ⊗[R] M₂) := (toEndomorphism R L M₂ x - χ₂ • ↑1).lTensor M₁
  have h_comm_square : F ∘ₗ ↑g = (g : M₁ ⊗[R] M₂ →ₗ[R] M₃).comp (f₁ + f₂) := by
    ext m₁ m₂;
    simp only [← g.map_lie x (m₁ ⊗ₜ m₂), add_smul, sub_tmul, tmul_sub, smul_tmul, lie_tmul_right,
      tmul_smul, toEndomorphism_apply_apply, LieModuleHom.map_smul, LinearMap.one_apply,
      LieModuleHom.coe_toLinearMap, LinearMap.smul_apply, Function.comp_apply, LinearMap.coe_comp,
      LinearMap.rTensor_tmul, LieModuleHom.map_add, LinearMap.add_apply, LieModuleHom.map_sub,
      LinearMap.sub_apply, LinearMap.lTensor_tmul, AlgebraTensorModule.curry_apply,
      TensorProduct.curry_apply, LinearMap.toFun_eq_coe, LinearMap.coe_restrictScalars]
    abel
  rsuffices ⟨k, hk⟩ : ∃ k : ℕ, ((f₁ + f₂) ^ k) (m₁ ⊗ₜ m₂) = 0
  · use k
    change (F ^ k) (g.toLinearMap (m₁ ⊗ₜ[R] m₂)) = 0
    rw [← LinearMap.comp_apply, LinearMap.commute_pow_left_of_commute h_comm_square,
      LinearMap.comp_apply, hk, LinearMap.map_zero]
  -- Unpack the information we have about `m₁`, `m₂`.
  simp only [Module.End.mem_maximalGeneralizedEigenspace] at hm₁ hm₂
  obtain ⟨k₁, hk₁⟩ := hm₁
  obtain ⟨k₂, hk₂⟩ := hm₂
  have hf₁ : (f₁ ^ k₁) (m₁ ⊗ₜ m₂) = 0 := by
    simp only [hk₁, zero_tmul, LinearMap.rTensor_tmul, LinearMap.rTensor_pow]
  have hf₂ : (f₂ ^ k₂) (m₁ ⊗ₜ m₂) = 0 := by
    simp only [hk₂, tmul_zero, LinearMap.lTensor_tmul, LinearMap.lTensor_pow]
  -- It's now just an application of the binomial theorem.
  use k₁ + k₂ - 1
  have hf_comm : Commute f₁ f₂ := by
    ext m₁ m₂
    simp only [LinearMap.mul_apply, LinearMap.rTensor_tmul, LinearMap.lTensor_tmul,
      AlgebraTensorModule.curry_apply, LinearMap.toFun_eq_coe, LinearMap.lTensor_tmul,
      TensorProduct.curry_apply, LinearMap.coe_restrictScalars]
  rw [hf_comm.add_pow']
  simp only [TensorProduct.mapIncl, Submodule.subtype_apply, Finset.sum_apply, Submodule.coe_mk,
    LinearMap.coeFn_sum, TensorProduct.map_tmul, LinearMap.smul_apply]
  -- The required sum is zero because each individual term is zero.
  

Proof state:
case h
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁹ : CommRing R
inst✝¹⁸ : LieRing L
inst✝¹⁷ : LieAlgebra R L
inst✝¹⁶ : LieAlgebra.IsNilpotent R L
inst✝¹⁵ : AddCommGroup M
inst✝¹⁴ : Module R M
inst✝¹³ : LieRingModule L M
inst✝¹² : LieModule R L M
M₁ : Type u_5
M₂ : Type u_6
M₃ : Type u_7
inst✝¹¹ : AddCommGroup M₁
inst✝¹⁰ : Module R M₁
inst✝⁹ : LieRingModule L M₁
inst✝⁸ : LieModule R L M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₂
inst✝⁵ : LieRingModule L M₂
inst✝⁴ : LieModule R L M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M₃
inst✝¹ : LieRingModule L M₃
inst✝ : LieModule R L M₃
g : M₁ ⊗[R] M₂ →ₗ⁅R,L⁆ M₃
χ₁ χ₂ : R
x : L
t : ↥𝕎(M₁, χ₁, x) ⊗[R] ↥𝕎(M₂, χ₂, x)
F : Module.End R M₃ := (toEndomorphism R L M₃) x - (χ₁ + χ₂) • 1
m₁ : M₁
m₂ : M₂
f₁ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.rTensor M₂ ((toEndomorphism R L M₁) x - χ₁ • 1)
f₂ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.lTensor M₁ ((toEndomorphism R L M₂) x - χ₂ • 1)
h_comm_square : F ∘ₗ ↑g = ↑g ∘ₗ (f₁ + f₂)
k₁ : ℕ
hk₁ : (((toEndomorphism R L M₁) x - χ₁ • 1) ^ k₁) m₁ = 0
k₂ : ℕ
hk₂ : (((toEndomorphism R L M₂) x - χ₂ • 1) ^ k₂) m₂ = 0
hf₁ : (f₁ ^ k₁) (m₁ ⊗ₜ[R] m₂) = 0
hf₂ : (f₂ ^ k₂) (m₁ ⊗ₜ[R] m₂) = 0
hf_comm : Commute f₁ f₂
⊢ ∑ x_1 in Finset.antidiagonal (k₁ + k₂ - 1),
      Nat.choose (k₁ + k₂ - 1) x_1.1 •
        (LinearMap.rTensor M₂ ((toEndomorphism R L M₁) x - χ₁ • 1) ^ x_1.1 *
            LinearMap.lTensor M₁ ((toEndomorphism R L M₂) x - χ₂ • 1) ^ x_1.2)
          (m₁ ⊗ₜ[R] m₂) =
    0

Next tactic:
apply Finset.sum_eq_zero