A : Type u_1
inst✝¹ : NonUnitalNonAssocRing A
inst✝ : IsCommJordan A
a b c : A
⊢ 0 = 2 • (⁅↑L a, ↑L (b * c)⁆ + ⁅↑L b, ↑L (c * a)⁆ + ⁅↑L c, ↑L (a * b)⁆)
A : Type u_1
inst✝¹ : NonUnitalNonAssocRing A
inst✝ : IsCommJordan A
a b c : A
⊢ 2 • (⁅↑L a, ↑L (b * c)⁆ + ⁅↑L b, ↑L (c * a)⁆ + ⁅↑L c, ↑L (a * b)⁆) = 0
symm
no goals
A : Type u_1
inst✝¹ : NonUnitalNonAssocRing A
inst✝ : IsCommJordan A
a b c : A
⊢ 0 = 2 • (⁅↑L a, ↑L (b * c)⁆ + ⁅↑L b, ↑L (c * a)⁆ + ⁅↑L c, ↑L (a * b)⁆)
calc
  0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by
    rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
  _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]
no goals
A : Type u_1
inst✝¹ : NonUnitalNonAssocRing A
inst✝ : IsCommJordan A
a b c : A
⊢ 0 = ⁅↑L (a + b + c), ↑L ((a + b + c) * (a + b + c))⁆
rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]
no goals
A : Type u_1
inst✝¹ : NonUnitalNonAssocRing A
inst✝ : IsCommJordan A
a b c : A
⊢ ⁅↑L (a + b + c), ↑L ((a + b + c) * (a + b + c))⁆ = 2 • (⁅↑L a, ↑L (b * c)⁆ + ⁅↑L b, ↑L (c * a)⁆ + ⁅↑L c, ↑L (a * b)⁆)
rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]