/-- The norm on `Unitization 𝕜 E` satisfies the C⋆-property -/
instance Unitization.instCstarRing : CstarRing (Unitization 𝕜 E) where
  norm_star_mul_self {x} := by
    -- rewrite both sides as a `⊔`
    simp only [Unitization.norm_def, Prod.norm_def, ← sup_eq_max]
    -- Show that `(Unitization.splitMul 𝕜 E x).snd` satisifes the C⋆-property, in two stages:
    have h₁ : ∀ x : Unitization 𝕜 E,
        ‖(Unitization.splitMul 𝕜 E x).snd‖ ≤ ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ := by
      simp only [add_zero, Unitization.splitMul_apply, Unitization.snd_star, Unitization.fst_star]
      intro x
      /- split based on whether the term inside the norm is zero or not. If so, it's trivial.
      If not, then apply `norm_splitMul_snd_sq` and cancel one copy of the norm -/
      by_cases h : algebraMap 𝕜 (E →L[𝕜] E) x.fst + mul 𝕜 E x.snd = 0
      · simp only [h, norm_zero, norm_le_zero_iff]
        exact norm_nonneg _
      · have : ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤
          ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ * ‖(Unitization.splitMul 𝕜 E x).snd‖ :=
          (norm_splitMul_snd_sq 𝕜 x).trans <| by
            rw [map_mul, Prod.snd_mul]
            exact norm_mul_le _ _
        rw [sq] at this
        rw [← Ne.def, ← norm_pos_iff] at h
        simp only [add_zero, Unitization.splitMul_apply, Unitization.snd_star,
          Unitization.fst_star, star_star] at this
        exact (mul_le_mul_right h).mp this
    -- in this step we make use of the key lemma `norm_splitMul_snd_sq`
    have h₂ : ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖
        = ‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 := by
      refine le_antisymm ?_ (norm_splitMul_snd_sq 𝕜 x)
      rw [map_mul, Prod.snd_mul]
      exact (norm_mul_le _ _).trans <| by
        rw [sq]
        gcongr
        simpa only [star_star] using h₁ (star x)
    -- Show that `(Unitization.splitMul 𝕜 E x).fst` satisfies the C⋆-property
    have h₃ : ‖(Unitization.splitMul 𝕜 E (star x * x)).fst‖
        = ‖(Unitization.splitMul 𝕜 E x).fst‖ ^ 2 := by
      simp only [Unitization.splitMul_apply, Unitization.fst_mul, Unitization.fst_star, add_zero,
        norm_mul, norm_star, sq]
    rw [h₂, h₃]
    /- use the definition of the norm, and split into cases based on whether the norm in the first
    coordinate is bigger or smaller than the norm in the second coordinate. -/
    by_cases h : ‖(Unitization.splitMul 𝕜 E x).fst‖ ≤ ‖(Unitization.splitMul 𝕜 E x).snd‖
    · 

Proof state:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CstarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : CstarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
h₁ : ∀ (x : Unitization 𝕜 E), ‖((splitMul 𝕜 E) x).2‖ ≤ ‖((splitMul 𝕜 E) x⋆).2‖
h₂ : ‖((splitMul 𝕜 E) (x⋆ * x)).2‖ = ‖((splitMul 𝕜 E) x).2‖ ^ 2
h₃ : ‖((splitMul 𝕜 E) (x⋆ * x)).1‖ = ‖((splitMul 𝕜 E) x).1‖ ^ 2
h : ‖((splitMul 𝕜 E) x).1‖ ≤ ‖((splitMul 𝕜 E) x).2‖
⊢ ‖((splitMul 𝕜 E) x).1‖ ^ 2 ⊔ ‖((splitMul 𝕜 E) x).2‖ ^ 2 =
    (‖((splitMul 𝕜 E) x).1‖ ⊔ ‖((splitMul 𝕜 E) x).2‖) * (‖((splitMul 𝕜 E) x).1‖ ⊔ ‖((splitMul 𝕜 E) x).2‖)

Next tactic:
rw [sq, sq, sup_eq_right.mpr h, sup_eq_right.mpr (mul_self_le_mul_self (norm_nonneg _) h)]