l3lab/ntp-mathlib huggingface dataset:
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z : L
⊢ traceForm R L M =
    ∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)

Next tactic:
ext x y
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
⊢ ((traceForm R L M) x) y =
    ((∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)) x) y

Next tactic:
have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
⊢ ((traceForm R L M) x) y =
    ((∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)) x) y

Next tactic:
have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
⊢ Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}

Next tactic:
convert finite_weightSpaceOf_ne_bot R L M z
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    

Proof state:
case h.e'_2.h.e'_2.h.h.a
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
x✝ : R
⊢ ↑(weightSpaceOf M x✝ z) = ⊥ ↔ weightSpaceOf M x✝ z = ⊥

Next tactic:
exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
hfin : Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}
⊢ ((traceForm R L M) x) y =
    ((∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)) x) y

Next tactic:
classical
  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)
    (IsTriangularizable.iSup_eq_top z)
  simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,
    LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]
  exact Finset.sum_congr (by simp) (fun χ _ ↦ rfl)
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
  classical
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
hfin : Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}
⊢ ((traceForm R L M) x) y =
    ((∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)) x) y

Next tactic:
have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)
    (IsTriangularizable.iSup_eq_top z)
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
  classical
  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)
    (IsTriangularizable.iSup_eq_top z)
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
hfin : Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}
hds : DirectSum.IsInternal fun i => ↑(weightSpaceOf M i z)
⊢ ((traceForm R L M) x) y =
    ((∑ χ in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}), traceForm R L ↥↑(weightSpaceOf M χ z)) x) y

Next tactic:
simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,
    LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
  classical
  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)
    (IsTriangularizable.iSup_eq_top z)
  simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,
    LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]
  

Proof state:
case h.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
hfin : Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}
hds : DirectSum.IsInternal fun i => ↑(weightSpaceOf M i z)
⊢ ∑ i in Finite.toFinset hfin,
      (trace R ↥↑(weightSpaceOf M i z))
        (LinearMap.restrict (φ x ∘ₗ φ y) (_ : MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M i z) ↑(weightSpaceOf M i z))) =
    ∑ x_1 in Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥}),
      (trace R ↥↑(weightSpaceOf M x_1 z))
        ((toEndomorphism R L ↥↑(weightSpaceOf M x_1 z)) x ∘ₗ (toEndomorphism R L ↥↑(weightSpaceOf M x_1 z)) y)

Next tactic:
exact Finset.sum_congr (by simp) (fun χ _ ↦ rfl)
lemma traceForm_eq_sum_weightSpaceOf [IsTriangularizable R L M] (z : L) :
    traceForm R L M =
    ∑ χ in (finite_weightSpaceOf_ne_bot R L M z).toFinset, traceForm R L (weightSpaceOf M χ z) := by
  ext x y
  have hxy : ∀ χ : R, MapsTo ((toEndomorphism R L M x).comp (toEndomorphism R L M y))
      (weightSpaceOf M χ z) (weightSpaceOf M χ z) :=
    fun χ m hm ↦ LieSubmodule.lie_mem _ <| LieSubmodule.lie_mem _ hm
  have hfin : {χ : R | (weightSpaceOf M χ z : Submodule R M) ≠ ⊥}.Finite := by
    convert finite_weightSpaceOf_ne_bot R L M z
    exact LieSubmodule.coeSubmodule_eq_bot_iff (weightSpaceOf M _ _)
  classical
  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_weightSpaceOf R L M z)
    (IsTriangularizable.iSup_eq_top z)
  simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,
    LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]
  exact Finset.sum_congr (by 

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁷ : CommRing R
inst✝¹⁶ : LieRing L
inst✝¹⁵ : LieAlgebra R L
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Module R M
inst✝¹² : LieRingModule L M
inst✝¹¹ : LieModule R L M
inst✝¹⁰ : Module.Free R M
inst✝⁹ : Module.Finite R M
inst✝⁸ : Field K
inst✝⁷ : LieAlgebra K L
inst✝⁶ : Module K M
inst✝⁵ : LieModule K L M
inst✝⁴ : FiniteDimensional K M
inst✝³ : LieAlgebra.IsNilpotent R L
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsTriangularizable R L M
z x y : L
hxy : ∀ (χ : R), MapsTo ⇑(φ x ∘ₗ φ y) ↑(weightSpaceOf M χ z) ↑(weightSpaceOf M χ z)
hfin : Set.Finite {χ | ↑(weightSpaceOf M χ z) ≠ ⊥}
hds : DirectSum.IsInternal fun i => ↑(weightSpaceOf M i z)
⊢ Finite.toFinset hfin = Finite.toFinset (_ : Set.Finite {χ | weightSpaceOf M χ z ≠ ⊥})

Next tactic:
simp
lemma lowerCentralSeries_one_inf_center_le_ker_traceForm :
    lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) := by
  /- Sketch of proof (due to Zassenhaus):

  Let `z ∈ lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L` and `x : L`. We must show that
  `trace (φ x ∘ φ z) = 0` where `φ z : End R M` indicates the action of `z` on `M` (and likewise
  for `φ x`).

  Because `z` belongs to the indicated intersection, it has two key properties:
  (a) the trace of the action of `z` vanishes on any Lie module of `L`
      (see `LieModule.trace_toEndomorphism_eq_zero_of_mem_lcs`),
  (b) `z` commutes with all elements of `L`.

  If `φ x` were triangularizable, we could write `M` as a direct sum of generalized eigenspaces of
  `φ x`. Because `L` is nilpotent these are all Lie submodules, thus Lie modules in their own right,
  and thus by (a) above we learn that `trace (φ z) = 0` restricted to each generalized eigenspace.
  Because `z` commutes with `x`, this forces `trace (φ x ∘ φ z) = 0` on each generalized eigenspace,
  and so by summing the traces on each generalized eigenspace we learn the total trace is zero, as
  required (see `LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero`).

  To cater for the fact that `φ x` may not be triangularizable, we first extend the scalars from `R`
  to `AlgebraicClosure (FractionRing R)` and argue using the action of `A ⊗ L` on `A ⊗ M`. -/
  

Proof state:
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁶ : CommRing R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : LieRingModule L M
inst✝¹⁰ : LieModule R L M
inst✝⁹ : Module.Free R M
inst✝⁸ : Module.Finite R M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieAlgebra.IsNilpotent R L
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
⊢ (lieIdealSubalgebra R L (lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L)).toSubmodule ≤
    LinearMap.ker (traceForm R L M)

Next tactic:
rintro z ⟨hz : z ∈ lowerCentralSeries R L L 1, hzc : z ∈ LieAlgebra.center R L⟩
lemma lowerCentralSeries_one_inf_center_le_ker_traceForm :
    lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) := by
  /- Sketch of proof (due to Zassenhaus):

  Let `z ∈ lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L` and `x : L`. We must show that
  `trace (φ x ∘ φ z) = 0` where `φ z : End R M` indicates the action of `z` on `M` (and likewise
  for `φ x`).

  Because `z` belongs to the indicated intersection, it has two key properties:
  (a) the trace of the action of `z` vanishes on any Lie module of `L`
      (see `LieModule.trace_toEndomorphism_eq_zero_of_mem_lcs`),
  (b) `z` commutes with all elements of `L`.

  If `φ x` were triangularizable, we could write `M` as a direct sum of generalized eigenspaces of
  `φ x`. Because `L` is nilpotent these are all Lie submodules, thus Lie modules in their own right,
  and thus by (a) above we learn that `trace (φ z) = 0` restricted to each generalized eigenspace.
  Because `z` commutes with `x`, this forces `trace (φ x ∘ φ z) = 0` on each generalized eigenspace,
  and so by summing the traces on each generalized eigenspace we learn the total trace is zero, as
  required (see `LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero`).

  To cater for the fact that `φ x` may not be triangularizable, we first extend the scalars from `R`
  to `AlgebraicClosure (FractionRing R)` and argue using the action of `A ⊗ L` on `A ⊗ M`. -/
  rintro z ⟨hz : z ∈ lowerCentralSeries R L L 1, hzc : z ∈ LieAlgebra.center R L⟩
  

Proof state:
case intro
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁶ : CommRing R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : LieRingModule L M
inst✝¹⁰ : LieModule R L M
inst✝⁹ : Module.Free R M
inst✝⁸ : Module.Finite R M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieAlgebra.IsNilpotent R L
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
z : L
hz : z ∈ lowerCentralSeries R L L 1
hzc : z ∈ LieAlgebra.center R L
⊢ z ∈ LinearMap.ker (traceForm R L M)

Next tactic:
ext x
lemma lowerCentralSeries_one_inf_center_le_ker_traceForm :
    lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) := by
  /- Sketch of proof (due to Zassenhaus):

  Let `z ∈ lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L` and `x : L`. We must show that
  `trace (φ x ∘ φ z) = 0` where `φ z : End R M` indicates the action of `z` on `M` (and likewise
  for `φ x`).

  Because `z` belongs to the indicated intersection, it has two key properties:
  (a) the trace of the action of `z` vanishes on any Lie module of `L`
      (see `LieModule.trace_toEndomorphism_eq_zero_of_mem_lcs`),
  (b) `z` commutes with all elements of `L`.

  If `φ x` were triangularizable, we could write `M` as a direct sum of generalized eigenspaces of
  `φ x`. Because `L` is nilpotent these are all Lie submodules, thus Lie modules in their own right,
  and thus by (a) above we learn that `trace (φ z) = 0` restricted to each generalized eigenspace.
  Because `z` commutes with `x`, this forces `trace (φ x ∘ φ z) = 0` on each generalized eigenspace,
  and so by summing the traces on each generalized eigenspace we learn the total trace is zero, as
  required (see `LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero`).

  To cater for the fact that `φ x` may not be triangularizable, we first extend the scalars from `R`
  to `AlgebraicClosure (FractionRing R)` and argue using the action of `A ⊗ L` on `A ⊗ M`. -/
  rintro z ⟨hz : z ∈ lowerCentralSeries R L L 1, hzc : z ∈ LieAlgebra.center R L⟩
  ext x
  

Proof state:
case intro.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁶ : CommRing R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : LieRingModule L M
inst✝¹⁰ : LieModule R L M
inst✝⁹ : Module.Free R M
inst✝⁸ : Module.Finite R M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieAlgebra.IsNilpotent R L
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
z : L
hz : z ∈ lowerCentralSeries R L L 1
hzc : z ∈ LieAlgebra.center R L
x : L
⊢ ((traceForm R L M) z) x = 0 x

Next tactic:
rw [traceForm_apply_apply, LinearMap.zero_apply]
lemma lowerCentralSeries_one_inf_center_le_ker_traceForm :
    lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) := by
  /- Sketch of proof (due to Zassenhaus):

  Let `z ∈ lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L` and `x : L`. We must show that
  `trace (φ x ∘ φ z) = 0` where `φ z : End R M` indicates the action of `z` on `M` (and likewise
  for `φ x`).

  Because `z` belongs to the indicated intersection, it has two key properties:
  (a) the trace of the action of `z` vanishes on any Lie module of `L`
      (see `LieModule.trace_toEndomorphism_eq_zero_of_mem_lcs`),
  (b) `z` commutes with all elements of `L`.

  If `φ x` were triangularizable, we could write `M` as a direct sum of generalized eigenspaces of
  `φ x`. Because `L` is nilpotent these are all Lie submodules, thus Lie modules in their own right,
  and thus by (a) above we learn that `trace (φ z) = 0` restricted to each generalized eigenspace.
  Because `z` commutes with `x`, this forces `trace (φ x ∘ φ z) = 0` on each generalized eigenspace,
  and so by summing the traces on each generalized eigenspace we learn the total trace is zero, as
  required (see `LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero`).

  To cater for the fact that `φ x` may not be triangularizable, we first extend the scalars from `R`
  to `AlgebraicClosure (FractionRing R)` and argue using the action of `A ⊗ L` on `A ⊗ M`. -/
  rintro z ⟨hz : z ∈ lowerCentralSeries R L L 1, hzc : z ∈ LieAlgebra.center R L⟩
  ext x
  rw [traceForm_apply_apply, LinearMap.zero_apply]
  

Proof state:
case intro.h
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁶ : CommRing R
inst✝¹⁵ : LieRing L
inst✝¹⁴ : LieAlgebra R L
inst✝¹³ : AddCommGroup M
inst✝¹² : Module R M
inst✝¹¹ : LieRingModule L M
inst✝¹⁰ : LieModule R L M
inst✝⁹ : Module.Free R M
inst✝⁸ : Module.Finite R M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieAlgebra.IsNilpotent R L
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
z : L
hz : z ∈ lowerCentralSeries R L L 1
hzc : z ∈ LieAlgebra.center R L
x : L
⊢ (trace R M) (φ z ∘ₗ φ x) = 0

Next tactic:
let A := AlgebraicClosure (FractionRing R)