-- Mathlib/Algebra/Jordan/Basic.lean
theorem aux1 {a b c : A} :
    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +
    2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆
    =
    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +
    ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +
    (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +
    ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +
    (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +
    ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆)

Proof state:
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =
    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
          ⁅L a, 2 • L (b * c)⁆ +
        (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
          ⁅L b, 2 • L (b * c)⁆) +
      (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
        ⁅L c, 2 • L (b * c)⁆)

Tactic:
rw [add_lie, add_lie]

Result:
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
        ⁅L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
      ⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =
    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
          ⁅L a, 2 • L (b * c)⁆ +
        (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
          ⁅L b, 2 • L (b * c)⁆) +
      (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
        ⁅L c, 2 • L (b * c)⁆)

Proof state:
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
        ⁅L b, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ +
      ⁅L c, L (a * a) + L (b * b) + L (c * c) + 2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ =
    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
          ⁅L a, 2 • L (b * c)⁆ +
        (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
          ⁅L b, 2 • L (b * c)⁆) +
      (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
        ⁅L c, 2 • L (b * c)⁆)

Tactic:
iterate 15 rw [lie_add]

Result:
no goals

Proof state:
A : Type u_1
inst✝ : NonUnitalNonAssocCommRing A
a b c : A
⊢ ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
          ⁅L a, 2 • L (b * c)⁆ +
        (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
          ⁅L b, 2 • L (b * c)⁆) +
      (⁅L c, L (a * a) + L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
        ⁅L c, 2 • L (b * c)⁆) =
    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ +
          ⁅L a, 2 • L (b * c)⁆ +
        (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ +
          ⁅L b, 2 • L (b * c)⁆) +
      (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ +
        ⁅L c, 2 • L (b * c)⁆)

Tactic:
rw [lie_add]

Result:
no goals