diff --git a/HepLean/SpaceTime/LorentzGroup/Basic.lean b/HepLean/SpaceTime/LorentzGroup/Basic.lean
index 98a12f5..5adf6fb 100644
--- a/HepLean/SpaceTime/LorentzGroup/Basic.lean
+++ b/HepLean/SpaceTime/LorentzGroup/Basic.lean
@@ -167,6 +167,27 @@ lemma subtype_mul_inv : (Subtype.val Λ) * (Subtype.val Λ)⁻¹ = 1 := by
   rw [mul_inv_self Λ]
   simp only [lorentzGroupIsGroup_one_coe]
 
+@[simp]
+lemma mul_minkowskiMatrix_mul_transpose :
+    (Subtype.val Λ) * minkowskiMatrix * (Subtype.val Λ).transpose = minkowskiMatrix := by
+  have h2 := Λ.prop
+  rw [LorentzGroup.mem_iff_self_mul_dual] at h2
+  simp [minkowskiMetric.dual] at h2
+  symm
+  refine right_inv_eq_left_inv minkowskiMatrix.sq ?_
+  rw [← h2]
+  noncomm_ring
+
+@[simp]
+lemma transpose_mul_minkowskiMatrix_mul_self :
+    (Subtype.val Λ).transpose * minkowskiMatrix * (Subtype.val Λ) = minkowskiMatrix := by
+  have h2 := Λ.prop
+  rw [LorentzGroup.mem_iff_dual_mul_self] at h2
+  simp [minkowskiMetric.dual] at h2
+  refine right_inv_eq_left_inv ?_ minkowskiMatrix.sq
+  rw [← h2]
+  noncomm_ring
+
 /-- The transpose of a matrix in the Lorentz group is an element of the Lorentz group. -/
 def transpose (Λ : LorentzGroup d) : LorentzGroup d :=
   ⟨Λ.1ᵀ, mem_iff_transpose.mp Λ.2⟩
diff --git a/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean b/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
index 745565e..dbf3abb 100644
--- a/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
+++ b/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
@@ -81,12 +81,12 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
     | .down => LorentzVector.unitDown_rid
   metric μ :=
     match μ with
-    | realTensor.ColorType.up => asProdLorentzVector
-    | realTensor.ColorType.down => asProdCovariantLorentzVector
+    | realTensor.ColorType.up => asTenProd
+    | realTensor.ColorType.down => asCoTenProd
   metric_dual μ :=
     match μ with
-    | realTensor.ColorType.up => asProdLorentzVector_contr_asCovariantProdLorentzVector
-    | realTensor.ColorType.down => asProdCovariantLorentzVector_contr_asProdLorentzVector
+    | realTensor.ColorType.up => asTenProd_contr_asCoTenProd
+    | realTensor.ColorType.down => asCoTenProd_contr_asTenProd
 
 /-- The action of the Lorentz group on real Lorentz tensors. -/
 instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where
@@ -98,5 +98,8 @@ instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where
     match μ with
     | .up => TensorProduct.ext' (fun _ _ => LorentzVector.contrUpDown_invariant_lorentzAction)
     | .down => TensorProduct.ext' (fun _ _ => LorentzVector.contrDownUp_invariant_lorentzAction)
-
+  metric_inv μ g :=
+    match μ with
+    | .up => asTenProd_invariant g
+    | .down => asCoTenProd_invariant g
 end
diff --git a/HepLean/SpaceTime/LorentzTensor/RisingLowering.lean b/HepLean/SpaceTime/LorentzTensor/RisingLowering.lean
index ed869d9..7a95683 100644
--- a/HepLean/SpaceTime/LorentzTensor/RisingLowering.lean
+++ b/HepLean/SpaceTime/LorentzTensor/RisingLowering.lean
@@ -176,7 +176,7 @@ lemma dualizeAll_equivariant (g : G) : (𝓣.dualizeAll.toLinearMap) ∘ₗ (@re
       simp [map_add, add_tmul, hx]
       simp_all only [Function.comp_apply, LinearMap.coe_comp, LinearMap.id_coe, id_eq])
   intro rx fx
-  simp
+  simp only [PiTensorProduct.tprodCoeff_eq_smul_tprod, LinearMapClass.map_smul, rep_tprod]
   apply congrArg
   change (PiTensorProduct.map _) ((PiTensorProduct.tprod R) _) =
     (𝓣.rep g) ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) fx))
diff --git a/HepLean/SpaceTime/LorentzVector/Contraction.lean b/HepLean/SpaceTime/LorentzVector/Contraction.lean
index 21c6e3a..25c3585 100644
--- a/HepLean/SpaceTime/LorentzVector/Contraction.lean
+++ b/HepLean/SpaceTime/LorentzVector/Contraction.lean
@@ -175,18 +175,41 @@ open scoped minkowskiMatrix
 variable {d : ℕ}
 
 /-- The metric tensor as an element of `LorentzVector d ⊗[ℝ] LorentzVector d`. -/
-def asProdLorentzVector : LorentzVector d ⊗[ℝ] LorentzVector d :=
-  ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ)
+def asTenProd : LorentzVector d ⊗[ℝ] LorentzVector d :=
+  ∑ μ, ∑ ν, η μ ν • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis ν)
+
+lemma asTenProd_diag :
+    @asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) := by
+  simp only [asTenProd]
+  refine Finset.sum_congr rfl (fun μ _ => ?_)
+  rw [Finset.sum_eq_single μ]
+  intro ν _ hμν
+  rw [minkowskiMatrix.off_diag_zero hμν.symm]
+  simp only [zero_smul]
+  intro a
+  simp_all only
 
 /-- The metric tensor as an element of `CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d`. -/
-def asProdCovariantLorentzVector : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
-  ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ)
+def asCoTenProd : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
+  ∑ μ, ∑ ν, η μ ν • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν)
+
+lemma asCoTenProd_diag :
+    @asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
+      CovariantLorentzVector.stdBasis μ) := by
+  simp only [asCoTenProd]
+  refine Finset.sum_congr rfl (fun μ _ => ?_)
+  rw [Finset.sum_eq_single μ]
+  intro ν _ hμν
+  rw [minkowskiMatrix.off_diag_zero hμν.symm]
+  simp only [zero_smul]
+  intro a
+  simp_all only
 
 @[simp]
-lemma contrLeft_asProdLorentzVector (x : CovariantLorentzVector d) :
-    contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asProdLorentzVector) =
+lemma contrLeft_asTenProd (x : CovariantLorentzVector d) :
+    contrLeftAux contrDownUp (x ⊗ₜ[ℝ] asTenProd) =
     ∑ μ, ((η μ μ * x μ) • LorentzVector.stdBasis μ) := by
-  simp only [asProdLorentzVector]
+  simp only [asTenProd_diag]
   rw [tmul_sum]
   rw [map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
@@ -196,10 +219,10 @@ lemma contrLeft_asProdLorentzVector (x : CovariantLorentzVector d) :
   exact smul_smul (η μ μ) (x μ) (e μ)
 
 @[simp]
-lemma contrLeft_asProdCovariantLorentzVector (x : LorentzVector d) :
-    contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asProdCovariantLorentzVector) =
+lemma contrLeft_asCoTenProd (x : LorentzVector d) :
+    contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
     ∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) := by
-  simp only [asProdCovariantLorentzVector]
+  simp only [asCoTenProd_diag]
   rw [tmul_sum]
   rw [map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
@@ -209,17 +232,17 @@ lemma contrLeft_asProdCovariantLorentzVector (x : LorentzVector d) :
   exact smul_smul (η μ μ) (x μ) (CovariantLorentzVector.stdBasis μ)
 
 @[simp]
-lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
-    (contrMidAux (contrUpDown) (asProdLorentzVector ⊗ₜ[ℝ] asProdCovariantLorentzVector))
+lemma asTenProd_contr_asCoTenProd :
+    (contrMidAux (contrUpDown) (asTenProd ⊗ₜ[ℝ] asCoTenProd))
     = TensorProduct.comm ℝ _ _ (@unitUp d) := by
-  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdLorentzVector, LinearMap.coe_comp,
+  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asTenProd_diag, LinearMap.coe_comp,
     LinearEquiv.coe_coe, Function.comp_apply]
   rw [sum_tmul, map_sum, map_sum, unitUp]
   simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
     Finset.sum_singleton, map_add, comm_tmul, map_sum]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
   rw [← tmul_smul, TensorProduct.assoc_tmul]
-  simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asProdCovariantLorentzVector]
+  simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asCoTenProd]
   rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul]
   change (η μ μ * (η μ μ * e μ μ)) • e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ = _
   rw [LorentzVector.stdBasis]
@@ -236,10 +259,10 @@ lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
   exact fun a => False.elim (hμν (id (Eq.symm a)))
 
 @[simp]
-lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
-    (contrMidAux (contrDownUp) (asProdCovariantLorentzVector ⊗ₜ[ℝ] asProdLorentzVector))
+lemma asCoTenProd_contr_asTenProd :
+    (contrMidAux (contrDownUp) (asCoTenProd ⊗ₜ[ℝ] asTenProd))
     = TensorProduct.comm ℝ _ _ (@unitDown d) := by
-  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asProdCovariantLorentzVector,
+  simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asCoTenProd_diag,
     LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
   rw [sum_tmul, map_sum, map_sum, unitDown]
   simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
@@ -247,7 +270,7 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
   refine Finset.sum_congr rfl (fun μ _ => ?_)
   rw [smul_tmul, TensorProduct.assoc_tmul]
   simp only [tmul_smul, LinearMapClass.map_smul, map_tmul, LinearMap.id_coe, id_eq,
-    contrLeft_asProdLorentzVector]
+    contrLeft_asTenProd]
   rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul, smul_smul, LorentzVector.stdBasis]
   erw [Pi.basisFun_apply]
   simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
@@ -260,6 +283,86 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
     ite_eq_right_iff, smul_eq_zero, mul_eq_zero]
   exact fun a => False.elim (hμν (id (Eq.symm a)))
 
+lemma asTenProd_invariant (g : LorentzGroup d) :
+    TensorProduct.map (LorentzVector.rep g) (LorentzVector.rep g) asTenProd = asTenProd := by
+  simp only [asTenProd, map_sum, LinearMapClass.map_smul, map_tmul, rep_apply_stdBasis,
+    Matrix.transpose_apply]
+  trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,
+      η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] ∑ ρ : Fin 1 ⊕ Fin d, g.1 ρ ν • e ρ
+  refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))
+  rw [sum_tmul, Finset.smul_sum]
+  trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
+      (η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] (g.1 ρ ν • e ρ))
+  · refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ =>
+      Finset.sum_congr rfl (fun φ _ => ?_)))
+    rw [tmul_sum, Finset.smul_sum]
+  rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_comm)]
+  rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => Finset.sum_comm))]
+  rw [Finset.sum_comm]
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_comm)]
+  trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d, ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d,
+      ((g.1 φ μ * η μ ν * g.1 ρ ν) • (e φ) ⊗ₜ[ℝ] (e ρ))
+  · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
+    Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))))
+    rw [smul_tmul, tmul_smul, tmul_smul, smul_smul, smul_smul]
+    ring_nf
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
+    Finset.sum_congr rfl (fun μ _ => Finset.sum_smul.symm)))]
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => Finset.sum_smul.symm))]
+  trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
+    (((g.1 * minkowskiMatrix * g.1.transpose : Matrix _ _ _) φ ρ) • (e φ) ⊗ₜ[ℝ] (e ρ))
+  · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => ?_))
+    apply congrFun
+    apply congrArg
+    simp only [Matrix.mul_apply, Matrix.transpose_apply]
+    rw [Finset.sum_comm]
+    refine Finset.sum_congr rfl (fun μ _ => ?_)
+    rw [Finset.sum_mul]
+  simp
+
+open CovariantLorentzVector in
+lemma asCoTenProd_invariant (g : LorentzGroup d) :
+    TensorProduct.map (CovariantLorentzVector.rep g) (CovariantLorentzVector.rep g)
+      asCoTenProd = asCoTenProd := by
+  simp only [asCoTenProd, map_sum, LinearMapClass.map_smul, map_tmul,
+    CovariantLorentzVector.rep_apply_stdBasis, Matrix.transpose_apply]
+  trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,
+      η μ ν • (g⁻¹.1 μ φ • CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
+        ∑ ρ : Fin 1 ⊕ Fin d, g⁻¹.1 ν ρ • CovariantLorentzVector.stdBasis ρ
+  · refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))
+    rw [sum_tmul, Finset.smul_sum]
+  trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
+      (η μ ν • (g⁻¹.1 μ φ • CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
+      (g⁻¹.1 ν ρ • CovariantLorentzVector.stdBasis ρ))
+  · refine Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ =>
+      Finset.sum_congr rfl (fun φ _ => ?_)))
+    rw [tmul_sum, Finset.smul_sum]
+  rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_comm)]
+  rw [Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => Finset.sum_comm))]
+  rw [Finset.sum_comm]
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_comm)]
+  trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d, ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d,
+      ((g⁻¹.1 μ φ * η μ ν * g⁻¹.1 ν ρ) • (CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ]
+      (CovariantLorentzVector.stdBasis ρ))
+  · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
+    Finset.sum_congr rfl (fun μ _ => Finset.sum_congr rfl (fun ν _ => ?_))))
+    rw [smul_tmul, tmul_smul, tmul_smul, smul_smul, smul_smul]
+    ring_nf
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ =>
+    Finset.sum_congr rfl (fun μ _ => Finset.sum_smul.symm)))]
+  rw [Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => Finset.sum_smul.symm))]
+  trans ∑ φ : Fin 1 ⊕ Fin d, ∑ ρ : Fin 1 ⊕ Fin d,
+    (((g⁻¹.1.transpose * minkowskiMatrix * g⁻¹.1 : Matrix _ _ _) φ ρ) •
+    (CovariantLorentzVector.stdBasis φ) ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis ρ))
+  · refine Finset.sum_congr rfl (fun φ _ => Finset.sum_congr rfl (fun ρ _ => ?_))
+    apply congrFun
+    apply congrArg
+    simp only [Matrix.mul_apply, Matrix.transpose_apply]
+    rw [Finset.sum_comm]
+    refine Finset.sum_congr rfl (fun μ _ => ?_)
+    rw [Finset.sum_mul]
+  rw [LorentzGroup.transpose_mul_minkowskiMatrix_mul_self]
+
 end minkowskiMatrix
 
 end
diff --git a/HepLean/SpaceTime/LorentzVector/Covariant.lean b/HepLean/SpaceTime/LorentzVector/Covariant.lean
index b2e799b..0deadf7 100644
--- a/HepLean/SpaceTime/LorentzVector/Covariant.lean
+++ b/HepLean/SpaceTime/LorentzVector/Covariant.lean
@@ -94,6 +94,16 @@ lemma rep_apply (g : LorentzGroup d) : rep g v = (g.1⁻¹)ᵀ *ᵥ v := by
   rw [← LorentzGroup.coe_inv]
   rfl
 
+lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
+    rep g (stdBasis μ) = ∑ ν, g⁻¹.1 μ ν • stdBasis ν := by
+  simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
+    Finset.sum_singleton, decomp_stdBasis']
+  funext ν
+  simp [LorentzVector.stdBasis, Pi.basisFun_apply]
+  erw [Pi.basisFun_apply, Matrix.mulVec_stdBasis]
+  rw [← LorentzGroup.coe_inv]
+  simp
+
 end CovariantLorentzVector
 
 end
diff --git a/HepLean/SpaceTime/LorentzVector/LorentzAction.lean b/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
index 40ad4d1..4a58bfd 100644
--- a/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
+++ b/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
@@ -28,6 +28,14 @@ def rep : Representation ℝ (LorentzGroup d) (LorentzVector d) where
 open Matrix in
 lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v := rfl
 
+lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
+    rep g (stdBasis μ) = ∑ ν, g.1.transpose μ ν • stdBasis ν := by
+  simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
+    Finset.sum_singleton, decomp_stdBasis']
+  funext ν
+  simp [LorentzVector.stdBasis, Pi.basisFun_apply]
+  erw [Pi.basisFun_apply, Matrix.mulVec_stdBasis]
+
 end LorentzVector
 
 end
diff --git a/HepLean/SpaceTime/MinkowskiMetric.lean b/HepLean/SpaceTime/MinkowskiMetric.lean
index 82da1fd..972b85f 100644
--- a/HepLean/SpaceTime/MinkowskiMetric.lean
+++ b/HepLean/SpaceTime/MinkowskiMetric.lean
@@ -79,6 +79,12 @@ lemma as_block : @minkowskiMatrix d = (
   rw [← diagonal_neg]
   rfl
 
+@[simp]
+lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
+  simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
+  apply diagonal_apply_ne
+  exact h
+
 end minkowskiMatrix
 
 /-!