/-- *Engel's theorem*.

Note that this implies all traditional forms of Engel's theorem via
`LieModule.nontrivial_max_triv_of_isNilpotent`, `LieModule.isNilpotent_iff_forall`,
`LieAlgebra.isNilpotent_iff_forall`. -/
theorem LieAlgebra.isEngelian_of_isNoetherian [IsNoetherian R L] : LieAlgebra.IsEngelian R L := by
  intro M _i1 _i2 _i3 _i4 h
  rw [← isNilpotent_range_toEndomorphism_iff]
  let L' := (toEndomorphism R L M).range
  replace h : ∀ y : L', _root_.IsNilpotent (y : Module.End R M)
  · rintro ⟨-, ⟨y, rfl⟩⟩
    simp [h]
  change LieModule.IsNilpotent R L' M
  let s := {K : LieSubalgebra R L' | LieAlgebra.IsEngelian R K}
  have hs : s.Nonempty := ⟨⊥, LieAlgebra.isEngelian_of_subsingleton⟩
  suffices ⊤ ∈ s by
    rw [← isNilpotent_of_top_iff]
    apply this M
    simp [LieSubalgebra.toEndomorphism_eq, h]
  have : ∀ K ∈ s, K ≠ ⊤ → ∃ K' ∈ s, K < K' := by
    rintro K (hK₁ : LieAlgebra.IsEngelian R K) hK₂
    apply LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer hK₁
    apply lt_of_le_of_ne K.le_normalizer
    rw [Ne.def, eq_comm, K.normalizer_eq_self_iff, ← Ne.def, ←
      LieSubmodule.nontrivial_iff_ne_bot R K]
    have : Nontrivial (L' ⧸ K.toLieSubmodule) := by
      replace hK₂ : K.toLieSubmodule ≠ ⊤ := by
        rwa [Ne.def, ← LieSubmodule.coe_toSubmodule_eq_iff, K.coe_toLieSubmodule,
          LieSubmodule.top_coeSubmodule, ← LieSubalgebra.top_coe_submodule,
          K.coe_to_submodule_eq_iff]
      exact Submodule.Quotient.nontrivial_of_lt_top _ hK₂.lt_top
    have : LieModule.IsNilpotent R K (L' ⧸ K.toLieSubmodule) := by
      -- Porting note: was refine' hK₁ _ fun x => _
      apply hK₁
      intro x
      have hx := LieAlgebra.isNilpotent_ad_of_isNilpotent (h x)
      apply Module.End.IsNilpotent.mapQ ?_ hx
      -- Porting note: mathlib3 solved this on its own with `submodule.mapq_linear._proof_5`
      intro X HX
      simp only [LieSubalgebra.coe_toLieSubmodule, LieSubalgebra.mem_coe_submodule] at HX
      simp only [LieSubalgebra.coe_toLieSubmodule, Submodule.mem_comap, ad_apply,
        LieSubalgebra.mem_coe_submodule]
      exact LieSubalgebra.lie_mem K x.prop HX
    exact nontrivial_max_triv_of_isNilpotent R K (L' ⧸ K.toLieSubmodule)
  haveI _i5 : IsNoetherian R L' := by
    -- Porting note: was
    -- isNoetherian_of_surjective L _ (LinearMap.range_rangeRestrict (toEndomorphism R L M))
    -- abusing the relation between `LieHom.rangeRestrict` and `LinearMap.rangeRestrict`
    refine isNoetherian_of_surjective L (LieHom.rangeRestrict (toEndomorphism R L M)) ?_
    simp only [LieHom.range_coeSubmodule, LieHom.coe_toLinearMap, LinearMap.range_eq_top]
    exact LieHom.surjective_rangeRestrict (toEndomorphism R L M)
  obtain ⟨K, hK₁, hK₂⟩ := (LieSubalgebra.wellFounded_of_noetherian R L').has_min s hs
  have hK₃ : K = ⊤ := by
    by_contra contra
    obtain ⟨K', hK'₁, hK'₂⟩ := this K hK₁ contra
    exact hK₂ K' hK'₁ hK'₂
  

Proof state:
case intro.intro
R : Type u₁
L : Type u₂
L₂ : Type u₃
M✝ : Type u₄
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R L
inst✝⁶ : LieRing L₂
inst✝⁵ : LieAlgebra R L₂
inst✝⁴ : AddCommGroup M✝
inst✝³ : Module R M✝
inst✝² : LieRingModule L M✝
inst✝¹ : LieModule R L M✝
inst✝ : IsNoetherian R L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
L' : LieSubalgebra R (Module.End R M) := LieHom.range (toEndomorphism R L M)
h : ∀ (y : ↥L'), _root_.IsNilpotent ↑y
s : Set (LieSubalgebra R ↥L') := {K | IsEngelian R ↥K}
hs : Set.Nonempty s
this : ∀ K ∈ s, K ≠ ⊤ → ∃ K' ∈ s, K < K'
_i5 : IsNoetherian R ↥L'
K : LieSubalgebra R ↥L'
hK₁ : K ∈ s
hK₂ : ∀ x ∈ s, ¬x > K
hK₃ : K = ⊤
⊢ ⊤ ∈ s

Next tactic:
exact hK₃ ▸ hK₁