/-- Given a family of Lie algebras `L i`, together with a family of morphisms of Lie algebras
`f i : L i →ₗ⁅R⁆ L'` into a fixed Lie algebra `L'`, we have a natural linear map:
`(⨁ i, L i) →ₗ[R] L'`. If in addition `⁅f i x, f j y⁆ = 0` for any `x ∈ L i` and `y ∈ L j` (`i ≠ j`)
then this map is a morphism of Lie algebras. -/
@[simps]
def toLieAlgebra [DecidableEq ι] (L' : Type w₁) [LieRing L'] [LieAlgebra R L']
    (f : ∀ i, L i →ₗ⁅R⁆ L') (hf : ∀ i j : ι, i ≠ j → ∀ (x : L i) (y : L j), ⁅f i x, f j y⁆ = 0) :
    (⨁ i, L i) →ₗ⁅R⁆ L' :=
  { toModule R ι L' fun i => (f i : L i →ₗ[R] L') with
    toFun := toModule R ι L' fun i => (f i : L i →ₗ[R] L')
    map_lie' := fun {x y} => by
      let f' i := (f i : L i →ₗ[R] L')
      /- The goal is linear in `y`. We can use this to reduce to the case that `y` has only one
        non-zero component. -/
      suffices ∀ (i : ι) (y : L i),
          toModule R ι L' f' ⁅x, of L i y⁆ =
            ⁅toModule R ι L' f' x, toModule R ι L' f' (of L i y)⁆ by
        

Proof state:
R : Type u
ι : Type v
inst✝⁵ : CommRing R
L : ι → Type w
inst✝⁴ : (i : ι) → LieRing (L i)
inst✝³ : (i : ι) → LieAlgebra R (L i)
inst✝² : DecidableEq ι
L' : Type w₁
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : (i : ι) → L i →ₗ⁅R⁆ L'
hf : ∀ (i j : ι), i ≠ j → ∀ (x : L i) (y : L j), ⁅(f i) x, (f j) y⁆ = 0
src✝ : (⨁ (i : ι), L i) →ₗ[R] L' := toModule R ι L' fun i => ↑(f i)
x y : ⨁ (i : ι), L i
f' : (i : ι) → L i →ₗ[R] L' := fun i => ↑(f i)
this :
  ∀ (i : ι) (y : L i),
    (toModule R ι L' f') ⁅x, (of L i) y⁆ = ⁅(toModule R ι L' f') x, (toModule R ι L' f') ((of L i) y)⁆
⊢ AddHom.toFun
      {
          toAddHom :=
            { toFun := ⇑(toModule R ι L' fun i => ↑(f i)),
              map_add' :=
                (_ :
                  ∀ (x y : ⨁ (i : ι), L i),
                    AddHom.toFun src✝.toAddHom (x + y) = AddHom.toFun src✝.toAddHom x + AddHom.toFun src✝.toAddHom y) },
          map_smul' :=
            (_ :
              ∀ (r : R) (x : ⨁ (i : ι), L i),
                AddHom.toFun src✝.toAddHom (r • x) = (RingHom.id R) r • AddHom.toFun src✝.toAddHom x) }.toAddHom
      ⁅x, y⁆ =
    ⁅AddHom.toFun
        {
            toAddHom :=
              { toFun := ⇑(toModule R ι L' fun i => ↑(f i)),
                map_add' :=
                  (_ :
                    ∀ (x y : ⨁ (i : ι), L i),
                      AddHom.toFun src✝.toAddHom (x + y) =
                        AddHom.toFun src✝.toAddHom x + AddHom.toFun src✝.toAddHom y) },
            map_smul' :=
              (_ :
                ∀ (r : R) (x : ⨁ (i : ι), L i),
                  AddHom.toFun src✝.toAddHom (r • x) = (RingHom.id R) r • AddHom.toFun src✝.toAddHom x) }.toAddHom
        x,
      AddHom.toFun
        {
            toAddHom :=
              { toFun := ⇑(toModule R ι L' fun i => ↑(f i)),
                map_add' :=
                  (_ :
                    ∀ (x y : ⨁ (i : ι), L i),
                      AddHom.toFun src✝.toAddHom (x + y) =
                        AddHom.toFun src✝.toAddHom x + AddHom.toFun src✝.toAddHom y) },
            map_smul' :=
              (_ :
                ∀ (r : R) (x : ⨁ (i : ι), L i),
                  AddHom.toFun src✝.toAddHom (r • x) = (RingHom.id R) r • AddHom.toFun src✝.toAddHom x) }.toAddHom
        y⁆

Next tactic:
simp only [← LieAlgebra.ad_apply R]