diff --git a/HepLean.lean b/HepLean.lean
index fb62704..f8931b7 100644
--- a/HepLean.lean
+++ b/HepLean.lean
@@ -73,11 +73,14 @@ import HepLean.SpaceTime.LorentzGroup.Rotations
 import HepLean.SpaceTime.LorentzTensor.Basic
 import HepLean.SpaceTime.LorentzTensor.Contractions
 import HepLean.SpaceTime.LorentzTensor.Fin
-import HepLean.SpaceTime.LorentzTensor.LorentzTensorStruct
+import HepLean.SpaceTime.LorentzTensor.MulActionTensor
 import HepLean.SpaceTime.LorentzTensor.Notation
-import HepLean.SpaceTime.LorentzTensor.RisingLowering
+import HepLean.SpaceTime.LorentzTensor.Real.Basic
 import HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix
 import HepLean.SpaceTime.LorentzVector.Basic
+import HepLean.SpaceTime.LorentzVector.Contraction
+import HepLean.SpaceTime.LorentzVector.Covariant
+import HepLean.SpaceTime.LorentzVector.LorentzAction
 import HepLean.SpaceTime.LorentzVector.NormOne
 import HepLean.SpaceTime.MinkowskiMetric
 import HepLean.SpaceTime.SL2C.Basic
diff --git a/HepLean/SpaceTime/LorentzGroup/Basic.lean b/HepLean/SpaceTime/LorentzGroup/Basic.lean
index fb1ecd9..98a12f5 100644
--- a/HepLean/SpaceTime/LorentzGroup/Basic.lean
+++ b/HepLean/SpaceTime/LorentzGroup/Basic.lean
@@ -151,6 +151,22 @@ lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= by
   refine (inv_eq_left_inv ?h).symm
   exact mem_iff_dual_mul_self.mp Λ.2
 
+@[simp]
+lemma subtype_inv_mul : (Subtype.val Λ)⁻¹ * (Subtype.val Λ) = 1 := by
+  trans Subtype.val (Λ⁻¹ * Λ)
+  rw [← coe_inv]
+  simp only [lorentzGroupIsGroup_inv, lorentzGroupIsGroup_mul_coe]
+  rw [inv_mul_self Λ]
+  simp only [lorentzGroupIsGroup_one_coe]
+
+@[simp]
+lemma subtype_mul_inv : (Subtype.val Λ) * (Subtype.val Λ)⁻¹ = 1 := by
+  trans Subtype.val (Λ * Λ⁻¹)
+  rw [← coe_inv]
+  simp only [lorentzGroupIsGroup_inv, lorentzGroupIsGroup_mul_coe]
+  rw [mul_inv_self Λ]
+  simp only [lorentzGroupIsGroup_one_coe]
+
 /-- 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/LorentzGroup/Boosts.lean b/HepLean/SpaceTime/LorentzGroup/Boosts.lean
index ea2f466..6f695b4 100644
--- a/HepLean/SpaceTime/LorentzGroup/Boosts.lean
+++ b/HepLean/SpaceTime/LorentzGroup/Boosts.lean
@@ -118,7 +118,7 @@ lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) := by
   simp only [toMatrix_apply]
   refine Continuous.comp' (continuous_mul_left (η i i)) ?hf
   refine Continuous.sub ?_ ?_
-  refine Continuous.comp' (continuous_add_left ⟪e i, e j⟫ₘ) ?_
+  refine Continuous.comp' (continuous_add_left (minkowskiMetric (e i) (e j))) ?_
   refine Continuous.comp' (continuous_mul_left (2 * ⟪e j, u⟫ₘ)) ?_
   exact FuturePointing.metric_continuous _
   refine Continuous.mul ?_ ?_
diff --git a/HepLean/SpaceTime/LorentzTensor/Basic.lean b/HepLean/SpaceTime/LorentzTensor/Basic.lean
index 468368c..62d3fd5 100644
--- a/HepLean/SpaceTime/LorentzTensor/Basic.lean
+++ b/HepLean/SpaceTime/LorentzTensor/Basic.lean
@@ -46,6 +46,7 @@ def contrDualMidAux {V1 V2 V3 V4 : Type} [AddCommMonoid V1] [AddCommMonoid V2] [
     (V4 ⊗[R] V1) ⊗[R] (V2 ⊗[R] V3) →ₗ[R] V4 ⊗[R] V3 :=
   (TensorProduct.map (LinearEquiv.refl R V4).toLinearMap (contrDualLeftAux f)) ∘ₗ
   (TensorProduct.assoc R _ _ _).toLinearMap
+
 /-- An initial structure specifying a tensor system (e.g. a system in which you can
   define real Lorentz tensors or Einstein notation convention). -/
 structure TensorStructure (R : Type) [CommSemiring R] where
@@ -72,9 +73,9 @@ structure TensorStructure (R : Type) [CommSemiring R] where
   /-- The unit is a right identity. -/
   unit_lid : ∀ μ (x : ColorModule μ),
     TensorProduct.rid R _
-    (TensorProduct.map  (LinearEquiv.refl R (ColorModule μ)).toLinearMap
-    (contrDual μ ∘ₗ (TensorProduct.comm R _ _).toLinearMap )
-    ((TensorProduct.assoc R _ _ _) (unit μ ⊗ₜ[R] x ))) = x
+    (TensorProduct.map (LinearEquiv.refl R (ColorModule μ)).toLinearMap
+    (contrDual μ ∘ₗ (TensorProduct.comm R _ _).toLinearMap)
+    ((TensorProduct.assoc R _ _ _) (unit μ ⊗ₜ[R] x))) = x
   /-- The metric for a given color. -/
   metric : (μ : Color) → ColorModule μ ⊗[R] ColorModule μ
   /-- The metric contracted with its dual is the unit. -/
diff --git a/HepLean/SpaceTime/LorentzTensor/LorentzTensorStruct.lean b/HepLean/SpaceTime/LorentzTensor/MulActionTensor.lean
similarity index 74%
rename from HepLean/SpaceTime/LorentzTensor/LorentzTensorStruct.lean
rename to HepLean/SpaceTime/LorentzTensor/MulActionTensor.lean
index af8f079..c2afbb5 100644
--- a/HepLean/SpaceTime/LorentzTensor/LorentzTensorStruct.lean
+++ b/HepLean/SpaceTime/LorentzTensor/MulActionTensor.lean
@@ -7,7 +7,7 @@ import HepLean.SpaceTime.LorentzTensor.Contractions
 import Mathlib.RepresentationTheory.Basic
 /-!
 
-# Structure to form Lorentz-style Tensor
+# Group actions on tensor structures
 
 -/
 
@@ -18,29 +18,65 @@ open TensorProduct
 variable {R : Type} [CommSemiring R]
 
 /-! TODO: Add preservation of the unit, and the metric. -/
-/-- A `DualizeTensorStructure` with a group action. -/
-structure LorentzTensorStructure (R : Type) [CommSemiring R]
-    (G : Type) [Group G] extends TensorStructure R where
+
+class MulActionTensor (G : Type) [Monoid G] (𝓣 : TensorStructure R) where
   /-- For each color `μ` a representation of `G` on `ColorModule μ`. -/
-  repColorModule : (μ : Color) → Representation R G (ColorModule μ)
+  repColorModule : (μ : 𝓣.Color) → Representation R G (𝓣.ColorModule μ)
   /-- The contraction of a vector with its dual is invariant under the group action. -/
-  contrDual_inv : ∀ μ g, contrDual μ ∘ₗ
-    TensorProduct.map (repColorModule μ g) (repColorModule (τ μ) g) = contrDual μ
+  contrDual_inv : ∀ μ g, 𝓣.contrDual μ ∘ₗ
+    TensorProduct.map (repColorModule μ g) (repColorModule (𝓣.τ μ) g) = 𝓣.contrDual μ
 
-namespace LorentzTensorStructure
-open TensorStructure
+namespace MulActionTensor
 
-variable {G : Type} [Group G]
+variable {G H : Type} [Group G] [Group H]
 
-variable (𝓣 : LorentzTensorStructure R G)
+variable (𝓣 : TensorStructure R) [MulActionTensor G 𝓣]
 
 variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
   [Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
   {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
 
+/-!
+
+# Instances of `MulActionTensor`
+
+-/
+
+def compHom (f : H →* G) : MulActionTensor H 𝓣 where
+  repColorModule μ := MonoidHom.comp (repColorModule μ) f
+  contrDual_inv μ h := by
+    simp only [MonoidHom.coe_comp, Function.comp_apply]
+    rw [contrDual_inv]
+
+def trivial : MulActionTensor G 𝓣 where
+  repColorModule μ := Representation.trivial R
+  contrDual_inv μ g := by
+    simp only [Representation.trivial, MonoidHom.one_apply, TensorProduct.map_one]
+    rfl
+
+end MulActionTensor
+
+namespace TensorStructure
+open TensorStructure
+open MulActionTensor
+
+variable {G : Type} [Group G]
+
+variable (𝓣 : TensorStructure R) [MulActionTensor G 𝓣]
+
+variable {d : ℕ} {X Y Y' Z : Type} [Fintype X] [DecidableEq X] [Fintype Y] [DecidableEq Y]
+  [Fintype Y'] [DecidableEq Y'] [Fintype Z] [DecidableEq Z]
+  {cX cX2 : X → 𝓣.Color} {cY : Y → 𝓣.Color} {cZ : Z → 𝓣.Color} {cY' : Y' → 𝓣.Color} {μ ν: 𝓣.Color}
+
+/-!
+
+## Representation of tensor products
+
+-/
+
 /-- The representation of the group `G` on the vector space of tensors. -/
 def rep : Representation R G (𝓣.Tensor cX) where
-  toFun g := PiTensorProduct.map (fun x => 𝓣.repColorModule (cX x) g)
+  toFun g := PiTensorProduct.map (fun x => repColorModule (cX x) g)
   map_one' := by
     simp_all only [_root_.map_one, PiTensorProduct.map_one]
   map_mul' g g' := by
@@ -50,15 +86,15 @@ def rep : Representation R G (𝓣.Tensor cX) where
 local infixl:78 " • " => 𝓣.rep
 
 lemma repColorModule_colorModuleCast_apply (h : μ = ν) (g : G) (x : 𝓣.ColorModule μ) :
-    (𝓣.repColorModule ν g) (𝓣.colorModuleCast h x) =
-    (𝓣.colorModuleCast h) (𝓣.repColorModule μ g x) := by
+    (repColorModule ν g) (𝓣.colorModuleCast h x) =
+    (𝓣.colorModuleCast h) (repColorModule μ g x) := by
   subst h
   simp [colorModuleCast]
 
 @[simp]
 lemma repColorModule_colorModuleCast (h : μ = ν) (g : G) :
-    (𝓣.repColorModule ν g) ∘ₗ (𝓣.colorModuleCast h).toLinearMap =
-    (𝓣.colorModuleCast h).toLinearMap ∘ₗ (𝓣.repColorModule μ g) := by
+    (repColorModule ν g) ∘ₗ (𝓣.colorModuleCast h).toLinearMap =
+    (𝓣.colorModuleCast h).toLinearMap ∘ₗ (repColorModule μ g) := by
   apply LinearMap.ext
   intro x
   simp [repColorModule_colorModuleCast_apply]
@@ -73,7 +109,7 @@ lemma rep_mapIso (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) :
     Function.comp_apply]
   erw [mapIso_tprod]
   simp [rep, repColorModule_colorModuleCast_apply]
-  change (PiTensorProduct.map fun x => (𝓣.repColorModule (cY x)) g)
+  change (PiTensorProduct.map fun x => (repColorModule (cY x)) g)
     ((PiTensorProduct.tprod R) fun i => (𝓣.colorModuleCast _) (x (e.symm i))) =
     (𝓣.mapIso e h) ((PiTensorProduct.map _) ((PiTensorProduct.tprod R) x))
   rw [PiTensorProduct.map_tprod, PiTensorProduct.map_tprod, mapIso_tprod]
@@ -92,7 +128,7 @@ lemma rep_mapIso_apply (e : X ≃ Y) (h : cX = cY ∘ e) (g : G) (x : 𝓣.Tenso
 @[simp]
 lemma rep_tprod (g : G) (f : (i : X) → 𝓣.ColorModule (cX i)) :
     g • (PiTensorProduct.tprod R f) = PiTensorProduct.tprod R (fun x =>
-    𝓣.repColorModule (cX x) g (f x)) := by
+    repColorModule (cX x) g (f x)) := by
   simp [rep]
   change (PiTensorProduct.map _) ((PiTensorProduct.tprod R) f) = _
   rw [PiTensorProduct.map_tprod]
@@ -163,7 +199,7 @@ lemma contrAll_rep {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (
   apply congrArg
   funext x
   rw [← repColorModule_colorModuleCast_apply]
-  nth_rewrite 2 [← 𝓣.contrDual_inv (c x) g]
+  nth_rewrite 2 [← contrDual_inv (c x) g]
   rfl
 
 @[simp]
@@ -177,9 +213,9 @@ lemma contrAll_rep_apply {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X 
 lemma contrAll_rep_tmul {c : X → 𝓣.Color} {d : Y → 𝓣.Color} (e : X ≃ Y) (h : c = 𝓣.τ ∘ d ∘ e)
     (g : G) (x : 𝓣.Tensor c) (y : 𝓣.Tensor d) :
     𝓣.contrAll e h ((g • x) ⊗ₜ[R] (g • y)) = 𝓣.contrAll e h (x ⊗ₜ[R] y) := by
-  nth_rewrite 2 [← contrAll_rep_apply]
+  nth_rewrite 2 [← @contrAll_rep_apply R _ G]
   rfl
 
-end LorentzTensorStructure
+end TensorStructure
 
 end
diff --git a/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean b/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
index 6ea904b..41bad45 100644
--- a/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
+++ b/HepLean/SpaceTime/LorentzTensor/Real/Basic.lean
@@ -4,7 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
 import HepLean.SpaceTime.LorentzVector.Contraction
-import HepLean.SpaceTime.LorentzTensor.Basic
+import HepLean.SpaceTime.LorentzVector.LorentzAction
+import HepLean.SpaceTime.LorentzTensor.MulActionTensor
 /-!
 
 # Real Lorentz tensors
@@ -85,4 +86,15 @@ def realLorentzTensor (d : ℕ) : TensorStructure ℝ where
       | realTensor.ColorType.up => asProdLorentzVector_contr_asCovariantProdLorentzVector
       | realTensor.ColorType.down => asProdCovariantLorentzVector_contr_asProdLorentzVector
 
+/-- The action of the Lorentz group on real Lorentz tensors. -/
+instance : MulActionTensor (LorentzGroup d) (realLorentzTensor d) where
+  repColorModule μ :=
+    match μ with
+    | .up => LorentzVector.rep
+    | .down => CovariantLorentzVector.rep
+  contrDual_inv μ _ :=
+    match μ with
+    | .up => TensorProduct.ext' (fun _ _ => LorentzVector.contrUpDown_invariant_lorentzAction)
+    | .down => TensorProduct.ext' (fun _ _ => LorentzVector.contrDownUp_invariant_lorentzAction)
+
 end
diff --git a/HepLean/SpaceTime/LorentzVector/Basic.lean b/HepLean/SpaceTime/LorentzVector/Basic.lean
index e0e2e10..9c4705c 100644
--- a/HepLean/SpaceTime/LorentzVector/Basic.lean
+++ b/HepLean/SpaceTime/LorentzVector/Basic.lean
@@ -86,7 +86,7 @@ lemma decomp_stdBasis (v : LorentzVector d) : ∑ i, v i • e i = v := by
 
 @[simp]
 lemma decomp_stdBasis' (v : LorentzVector d) :
-    v (Sum.inl 0) • e (Sum.inl 0) + ∑ a₂ : Fin d, v (Sum.inr a₂) • e (Sum.inr a₂)  = v := by
+    v (Sum.inl 0) • e (Sum.inl 0) + ∑ a₂ : Fin d, v (Sum.inr a₂) • e (Sum.inr a₂) = v := by
   trans ∑ i, v i • e i
   simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
     Finset.sum_singleton]
diff --git a/HepLean/SpaceTime/LorentzVector/Contraction.lean b/HepLean/SpaceTime/LorentzVector/Contraction.lean
index e410777..7238c20 100644
--- a/HepLean/SpaceTime/LorentzVector/Contraction.lean
+++ b/HepLean/SpaceTime/LorentzVector/Contraction.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Tooby-Smith
 -/
-import HepLean.SpaceTime.LorentzVector.Basic
+import HepLean.SpaceTime.LorentzVector.LorentzAction
 import HepLean.SpaceTime.LorentzVector.Covariant
 import HepLean.SpaceTime.LorentzTensor.Basic
 /-!
@@ -24,9 +24,9 @@ open TensorProduct
 
 namespace LorentzVector
 
+open Matrix
 variable {d : ℕ} (v : LorentzVector d)
 
-
 def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ where
   toFun v := {
     toFun := fun w => ∑ i, v i * w i,
@@ -60,6 +60,10 @@ def contrUpDownBi : LorentzVector d →ₗ[ℝ] CovariantLorentzVector d →ₗ[
 def contrUpDown : LorentzVector d ⊗[ℝ] CovariantLorentzVector d →ₗ[ℝ] ℝ :=
   TensorProduct.lift contrUpDownBi
 
+lemma contrUpDown_tmul_eq_dotProduct {x : LorentzVector d} {y : CovariantLorentzVector d} :
+    contrUpDown (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
+  rfl
+
 @[simp]
 lemma contrUpDown_stdBasis_left (x : LorentzVector d) (i : Fin 1 ⊕ Fin d) :
     contrUpDown (x ⊗ₜ[ℝ] (CovariantLorentzVector.stdBasis i)) = x i := by
@@ -92,16 +96,26 @@ lemma contrUpDown_stdBasis_right (x : CovariantLorentzVector d) (i : Fin 1 ⊕ F
 def contrDownUp : CovariantLorentzVector d ⊗[ℝ] LorentzVector d →ₗ[ℝ] ℝ :=
   contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap
 
+lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
+    contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y := by
+  rw [dotProduct_comm x y]
+  rfl
+
+/-!
+
+## The unit of the contraction
+
+-/
 
 def unitUp : LorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
   ∑ i, LorentzVector.stdBasis i ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis i
 
 lemma unitUp_lid (x : LorentzVector d) :
     TensorProduct.rid ℝ _
-    (TensorProduct.map  (LinearEquiv.refl ℝ (_)).toLinearMap
-    (contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap )
-    ((TensorProduct.assoc ℝ _ _ _) (unitUp ⊗ₜ[ℝ] x )))  = x := by
-  simp only  [LinearEquiv.refl_toLinearMap, unitUp]
+    (TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
+    (contrUpDown ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
+    ((TensorProduct.assoc ℝ _ _ _) (unitUp ⊗ₜ[ℝ] x))) = x := by
+  simp only [LinearEquiv.refl_toLinearMap, unitUp]
   rw [sum_tmul]
   simp only [Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
     Finset.sum_singleton, map_add, assoc_tmul, map_sum, map_tmul, LinearMap.id_coe, id_eq,
@@ -113,8 +127,8 @@ def unitDown : CovariantLorentzVector d ⊗[ℝ] LorentzVector d :=
 
 lemma unitDown_lid (x : CovariantLorentzVector d) :
     TensorProduct.rid ℝ _
-    (TensorProduct.map  (LinearEquiv.refl ℝ (_)).toLinearMap
-    (contrDownUp ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap )
+    (TensorProduct.map (LinearEquiv.refl ℝ _).toLinearMap
+    (contrDownUp ∘ₗ (TensorProduct.comm ℝ _ _).toLinearMap)
     ((TensorProduct.assoc ℝ _ _ _) (unitDown ⊗ₜ[ℝ] x))) = x := by
   simp only [LinearEquiv.refl_toLinearMap, unitDown]
   rw [sum_tmul]
@@ -123,6 +137,29 @@ lemma unitDown_lid (x : CovariantLorentzVector d) :
     id_eq, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, comm_tmul,
     contrUpDown_stdBasis_right, rid_tmul, CovariantLorentzVector.decomp_stdBasis']
 
+/-!
+
+# Contractions and the Lorentz actions
+
+-/
+open Matrix
+
+@[simp]
+lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
+    (CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) := by
+  rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]
+  simp only [rep_apply, CovariantLorentzVector.rep_apply]
+  rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
+  simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
+
+@[simp]
+lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
+    (LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) := by
+  rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]
+  rw [dotProduct_comm, dotProduct_comm x y]
+  simp only [rep_apply, CovariantLorentzVector.rep_apply]
+  rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]
+  simp only [LorentzGroup.subtype_inv_mul, one_mulVec]
 
 end LorentzVector
 
@@ -132,7 +169,7 @@ open scoped minkowskiMatrix
 variable {d : ℕ}
 
 def asProdLorentzVector : LorentzVector d ⊗[ℝ] LorentzVector d :=
-  ∑ μ,  η μ μ  • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ)
+  ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ)
 
 def asProdCovariantLorentzVector : CovariantLorentzVector d ⊗[ℝ] CovariantLorentzVector d :=
   ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ)
@@ -173,16 +210,15 @@ lemma asProdLorentzVector_contr_asCovariantProdLorentzVector :
   refine Finset.sum_congr rfl (fun μ _ => ?_)
   rw [← tmul_smul, TensorProduct.assoc_tmul]
   simp only [map_tmul, LinearMap.id_coe, id_eq, contrLeft_asProdCovariantLorentzVector]
-  rw [tmul_sum, Finset.sum_eq_single_of_mem μ]
-  rw [tmul_smul]
-  change (η μ μ * (η μ μ * e μ μ)) •  e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis μ = _
+  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]
   erw [Pi.basisFun_apply]
-  simp
+  simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
   exact Finset.mem_univ μ
   intro ν _ hμν
   rw [tmul_smul]
-  change  (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
+  change (η ν ν * (η μ μ * e μ ν)) • (e μ ⊗ₜ[ℝ] CovariantLorentzVector.stdBasis ν) = _
   rw [LorentzVector.stdBasis]
   erw [Pi.basisFun_apply]
   simp only [LinearMap.stdBasis_apply', mul_ite, mul_one, mul_zero, ite_smul, zero_smul,
@@ -197,15 +233,12 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
     LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
   rw [sum_tmul, map_sum, map_sum, unitDown]
   refine Finset.sum_congr rfl (fun μ _ => ?_)
-  rw [smul_tmul,]
-  rw [TensorProduct.assoc_tmul]
+  rw [smul_tmul, TensorProduct.assoc_tmul]
   simp only [tmul_smul, LinearMapClass.map_smul, map_tmul, LinearMap.id_coe, id_eq,
     contrLeft_asProdLorentzVector]
-  rw [tmul_sum, Finset.sum_eq_single_of_mem μ]
-  rw [tmul_smul, smul_smul]
-  rw [LorentzVector.stdBasis]
+  rw [tmul_sum, Finset.sum_eq_single_of_mem μ, tmul_smul, smul_smul, LorentzVector.stdBasis]
   erw [Pi.basisFun_apply]
-  simp
+  simp only [LinearMap.stdBasis_same, mul_one, η_apply_mul_η_apply_diag, one_smul]
   exact Finset.mem_univ μ
   intro ν _ hμν
   rw [tmul_smul]
@@ -217,5 +250,4 @@ lemma asProdCovariantLorentzVector_contr_asProdLorentzVector :
 
 end minkowskiMatrix
 
-
 end
diff --git a/HepLean/SpaceTime/LorentzVector/Covariant.lean b/HepLean/SpaceTime/LorentzVector/Covariant.lean
index 017b6cf..1e81d32 100644
--- a/HepLean/SpaceTime/LorentzVector/Covariant.lean
+++ b/HepLean/SpaceTime/LorentzVector/Covariant.lean
@@ -61,7 +61,7 @@ lemma decomp_stdBasis (v : CovariantLorentzVector d) : ∑ i, v i • stdBasis i
 
 @[simp]
 lemma decomp_stdBasis' (v : CovariantLorentzVector d) :
-    v (Sum.inl 0) • stdBasis (Sum.inl 0) + ∑ a₂ : Fin d, v (Sum.inr a₂) • stdBasis (Sum.inr a₂)  = v := by
+    v (Sum.inl 0) • stdBasis (Sum.inl 0) + ∑ a₂ : Fin d, v (Sum.inr a₂) • stdBasis (Sum.inr a₂) = v := by
   trans ∑ i, v i • stdBasis i
   simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
     Finset.sum_singleton]
@@ -82,6 +82,17 @@ def rep : Representation ℝ (LorentzGroup d) (CovariantLorentzVector d) where
     simp only [mul_inv_rev, lorentzGroupIsGroup_inv, LorentzGroup.transpose_mul,
       lorentzGroupIsGroup_mul_coe, map_mul]
 
+open Matrix in
+@[simp]
+lemma rep_apply (g : LorentzGroup d) : rep g v = (g.1⁻¹)ᵀ *ᵥ v := by
+  trans (LorentzGroup.transpose g⁻¹) *ᵥ v
+  rfl
+  apply congrFun
+  apply congrArg
+  simp only [LorentzGroup.transpose, lorentzGroupIsGroup_inv, transpose_inj]
+  rw [← LorentzGroup.coe_inv]
+  rfl
+
 end CovariantLorentzVector
 
 end
diff --git a/HepLean/SpaceTime/LorentzVector/LorentzAction.lean b/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
index cfbe46f..d4b0944 100644
--- a/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
+++ b/HepLean/SpaceTime/LorentzVector/LorentzAction.lean
@@ -25,6 +25,10 @@ def rep : Representation ℝ (LorentzGroup d) (LorentzVector d) where
   map_mul' x y := by
     simp only [lorentzGroupIsGroup_mul_coe, map_mul]
 
+open Matrix in
+@[simp]
+lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v := rfl
+
 end LorentzVector
 
 end
diff --git a/scripts/check_file_imports.lean b/scripts/check_file_imports.lean
index 394826c..cf7e1d5 100644
--- a/scripts/check_file_imports.lean
+++ b/scripts/check_file_imports.lean
@@ -68,10 +68,17 @@ def checkMissingImports (modData : ModuleData) (reqImports : Array Name) :
     IO Bool := do
   let names : HashSet Name := HashSet.ofArray (modData.imports.map (·.module))
   let mut warned := false
-  for req in reqImports do
-    if !names.contains req then
-      IO.print s!"File {req} is not imported. \n"
-      warned := true
+  let nameArray := reqImports.filterMap (
+    fun req => if !names.contains req then
+      some req
+    else
+      none)
+  if nameArray.size ≠ 0  then
+    let nameArraySort := nameArray.qsort (·.toString < ·.toString)
+    IO.print s!"Files are not imported add the following to the `HepLean` file: \n"
+    for name in nameArraySort do
+      IO.print s!"import {name}\n"
+    warned := true
   pure warned
 
 def main (_ : List String) : IO UInt32 := do