/-- A Lie module yields a Lie algebra morphism into the linear endomorphisms of the module.

See also `LieModule.toModuleHom`. -/
@[simps]
def LieModule.toEndomorphism : L →ₗ⁅R⁆ Module.End R M where
  toFun x :=
    { toFun := fun m => ⁅x, m⁆
      map_add' := lie_add x
      map_smul' := fun t => lie_smul t x }
  map_add' x y := by ext m; apply add_lie
  map_smul' t x := by 

Proof state:
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
t : R
x : L
⊢ AddHom.toFun
      {
        toFun := fun x =>
          { toAddHom := { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
            map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) },
        map_add' :=
          (_ :
            ∀ (x y : L),
              (fun x =>
                    {
                      toAddHom :=
                        { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                      map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                  (x + y) =
                (fun x =>
                      {
                        toAddHom :=
                          { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                        map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                    x +
                  (fun x =>
                      {
                        toAddHom :=
                          { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                        map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                    y) }
      (t • x) =
    (RingHom.id R) t •
      AddHom.toFun
        {
          toFun := fun x =>
            { toAddHom := { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
              map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) },
          map_add' :=
            (_ :
              ∀ (x y : L),
                (fun x =>
                      {
                        toAddHom :=
                          { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                        map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                    (x + y) =
                  (fun x =>
                        {
                          toAddHom :=
                            { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                          map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                      x +
                    (fun x =>
                        {
                          toAddHom :=
                            { toFun := fun m => ⁅x, m⁆, map_add' := (_ : ∀ (m n : M), ⁅x, m + n⁆ = ⁅x, m⁆ + ⁅x, n⁆) },
                          map_smul' := (_ : ∀ (t : R) (m : M), ⁅x, t • m⁆ = t • ⁅x, m⁆) })
                      y) }
        x

Next tactic:
ext m