{R : Type u_1} →
  {A : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : Ring A] →
        [inst_2 : Algebra R A] →
          {c₁ c₂ c₃ : R} →
            (i j k : A) →
              i * i = c₁ • 1 + c₂ • i →
                j * j = c₃ • 1 → i * j = k → j * i = c₂ • j - k → QuaternionAlgebra.Basis A c₁ c₂ c₃