-- Mathlib/Algebra/Lie/BaseChange.lean
theorem bracket_leibniz_lie (x y : A ⊗[R] L) (z : A ⊗[R] M) :
    ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆

Proof state:
case refine_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
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
x y : A ⊗[R] L
z : A ⊗[R] M
u₁ u₂ : A ⊗[R] L
h₁ :
  ((LieAlgebra.ExtendScalars.bracket' R A L M) u₁) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) u₁) y)) z +
      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) u₁) z)
h₂ :
  ((LieAlgebra.ExtendScalars.bracket' R A L M) u₂) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) u₂) y)) z +
      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) u₂) z)
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L M) (u₁ + u₂)) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (u₁ + u₂)) y)) z +
      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) (u₁ + u₂)) z)

Tactic:
rw [map_add, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, LinearMap.add_apply,
  map_add, LinearMap.add_apply, h₁, h₂, add_add_add_comm]

Result:
no goals

Proof state:
case refine_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
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
x y : A ⊗[R] L
z : A ⊗[R] M
a₁ : A
l₁ : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y)) z +
      ((LieAlgebra.ExtendScalars.bracket' R A L M) y) (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z)

Tactic:
refine y.induction_on ?_ ?_ ?_

Result:
case refine_2.refine_1
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
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
x y : A ⊗[R] L
z : A ⊗[R] M
a₁ : A
l₁ : L
⊢ ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) (((LieAlgebra.ExtendScalars.bracket' R A L M) 0) z) =
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) 0)) z +
      ((LieAlgebra.ExtendScalars.bracket' R A L M) 0) (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z)

case refine_2.refine_2
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
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
x y : A ⊗[R] L
z : A ⊗[R] M
a₁ : A
l₁ : L
⊢ ∀ (x : A) (y : L),
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁))
        (((LieAlgebra.ExtendScalars.bracket' R A L M) (x ⊗ₜ[R] y)) z) =
      ((LieAlgebra.ExtendScalars.bracket' R A L M)
            (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (x ⊗ₜ[R] y)))
          z +
        ((LieAlgebra.ExtendScalars.bracket' R A L M) (x ⊗ₜ[R] y))
          (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z)

case refine_2.refine_3
R : Type u_1
A : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
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
x y : A ⊗[R] L
z : A ⊗[R] M
a₁ : A
l₁ : L
⊢ ∀ (x y : A ⊗[R] L),
    ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) (((LieAlgebra.ExtendScalars.bracket' R A L M) x) z) =
        ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) x))
            z +
          ((LieAlgebra.ExtendScalars.bracket' R A L M) x)
            (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z) →
      ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) (((LieAlgebra.ExtendScalars.bracket' R A L M) y) z) =
          ((LieAlgebra.ExtendScalars.bracket' R A L M) (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) y))
              z +
            ((LieAlgebra.ExtendScalars.bracket' R A L M) y)
              (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z) →
        ((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁))
            (((LieAlgebra.ExtendScalars.bracket' R A L M) (x + y)) z) =
          ((LieAlgebra.ExtendScalars.bracket' R A L M)
                (((LieAlgebra.ExtendScalars.bracket' R A L L) (a₁ ⊗ₜ[R] l₁)) (x + y)))
              z +
            ((LieAlgebra.ExtendScalars.bracket' R A L M) (x + y))
              (((LieAlgebra.ExtendScalars.bracket' R A L M) (a₁ ⊗ₜ[R] l₁)) z)