∀ {L₁ : Type u_1} {L₂ : Type u_2} {M : Type u_3} [inst : Bracket L₁ L₂] [inst_1 : Bracket L₁ M] [inst_2 : Bracket L₂ M]
  [inst_3 : Add M], (∀ (x : L₁) (y : L₂) (m : M), ⁅x, ⁅y, m⁆⁆ = ⁅⁅x, y⁆, m⁆ + ⁅y, ⁅x, m⁆⁆) → IsLieTower L₁ L₂ M