diff --git a/HepLean/Tensors/ComplexLorentz/Basis.lean b/HepLean/Tensors/ComplexLorentz/Basis.lean
index 61f9d31..ec707a7 100644
--- a/HepLean/Tensors/ComplexLorentz/Basis.lean
+++ b/HepLean/Tensors/ComplexLorentz/Basis.lean
@@ -98,6 +98,34 @@ lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.
   erw [basis_contr]
   rfl
 
+lemma contr_basisVector_tree  {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
+    {i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
+    (contr i j h (tensorNode (basisVector c b))).tensor =
+    (smul ((if (b i).val = (b (i.succAbove j)).val
+    then (1 : ℂ) else 0)) (tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
+    (fun k => b (i.succAbove (j.succAbove k)))) )).tensor := by
+  exact contr_basisVector _
+
+lemma contr_basisVector_tree_pos  {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
+    {i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) (hn : (b i).val = (b (i.succAbove j)).val := by decide) :
+    (contr i j h (tensorNode (basisVector c b))).tensor =
+    ((tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
+    (fun k => b (i.succAbove (j.succAbove k)))))).tensor := by
+  rw [contr_basisVector_tree]
+  rw [if_pos hn]
+  simp [smul_tensor]
+
+lemma contr_basisVector_tree_neg  {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
+    {i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k))) (hn : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
+    (contr i j h (tensorNode (basisVector c b))).tensor =
+    (tensorNode 0).tensor := by
+  rw [contr_basisVector_tree]
+  rw [if_neg hn]
+  simp [smul_tensor]
+
 def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
     {c1 : Fin m → complexLorentzTensor.C} (i : Fin n ⊕ Fin m) :
     Sum.elim (fun i => Fin (complexLorentzTensor.repDim (c i))) (fun i => Fin (complexLorentzTensor.repDim (c1 i)))
@@ -135,6 +163,16 @@ lemma prod_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
   | Sum.inl k => rfl
   | Sum.inr k => rfl
 
+lemma prod_basisVector_tree  {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
+    {c1 : Fin m → complexLorentzTensor.C}
+    (b : Π k, Fin (complexLorentzTensor.repDim (c k)))
+    (b1 : Π k, Fin (complexLorentzTensor.repDim (c1 k))) :
+   (prod (tensorNode (basisVector c b)) (tensorNode (basisVector c1 b1))).tensor =
+    (tensorNode (basisVector (Sum.elim c c1 ∘ finSumFinEquiv.symm) (fun i =>
+    prodBasisVecEquiv (finSumFinEquiv.symm i)
+    ((HepLean.PiTensorProduct.elimPureTensor b b1) (finSumFinEquiv.symm i))))).tensor := by
+  exact prod_basisVector _ _
+
 lemma eval_basisVector  {n : ℕ} {c : Fin n.succ → complexLorentzTensor.C}
     {i : Fin n.succ} (j : Fin (complexLorentzTensor.repDim (c i)))
     (b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
@@ -183,6 +221,13 @@ lemma coMetric_basis_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
       simp only [Fin.isValue, Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
       rfl
 
+lemma coMetric_basis_expand_tree : {Lorentz.coMetric | μ ν}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 0))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 1)))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 2)))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.down, Color.down] (fun _ => 3))))).tensor :=
+  coMetric_basis_expand
+
 /-- The expansion of the Lorentz contrvariant metric in terms of basis vectors. -/
 lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
     basisVector ![Color.up, Color.up] (fun _ => 0)
@@ -208,6 +253,13 @@ lemma contrMatrix_basis_expand : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
       simp only [Fin.isValue, Lorentz.complexContrBasisFin4, Basis.coe_reindex, Function.comp_apply]
       rfl
 
+lemma contrMatrix_basis_expand_tree : {Lorentz.contrMetric | μ ν}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 0))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 1)))) <|
+    TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 2)))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.up, Color.up] (fun _ => 3))))).tensor :=
+  contrMatrix_basis_expand
+
 lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
    - basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)
     + basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0) := by
@@ -225,6 +277,11 @@ lemma leftMetric_expand : {Fermion.leftMetric | α β}ᵀ.tensor =
     · rfl
     · rfl
 
+lemma leftMetric_expand_tree : {Fermion.leftMetric | α β}ᵀ.tensor =
+    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 0 | 1 => 1)))) <|
+    (tensorNode (basisVector ![Color.upL, Color.upL] (fun | 0 => 1 | 1 => 0)))).tensor :=
+  leftMetric_expand
+
 lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
     basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1)
     - basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0) := by
@@ -241,6 +298,11 @@ lemma altLeftMetric_expand : {Fermion.altLeftMetric | α β}ᵀ.tensor =
     · rfl
     · rfl
 
+lemma altLeftMetric_expand_tree : {Fermion.altLeftMetric | α β}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.downL, Color.downL] (fun | 0 => 0 | 1 => 1))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.downL, Color.downL] (fun | 0 => 1 | 1 => 0))))).tensor :=
+  altLeftMetric_expand
+
 lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
     - basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)
     + basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0) := by
@@ -258,6 +320,11 @@ lemma rightMetric_expand : {Fermion.rightMetric | α β}ᵀ.tensor =
     · rfl
     · rfl
 
+lemma rightMetric_expand_tree : {Fermion.rightMetric | α β}ᵀ.tensor =
+    (TensorTree.add (smul (-1) (tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 0 | 1 => 1)))) <|
+    (tensorNode (basisVector ![Color.upR, Color.upR] (fun | 0 => 1 | 1 => 0)))).tensor :=
+  rightMetric_expand
+
 lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
     basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1)
     - basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0) := by
@@ -274,6 +341,11 @@ lemma altRightMetric_expand : {Fermion.altRightMetric | α β}ᵀ.tensor =
     · rfl
     · rfl
 
+lemma altRightMetric_expand_tree : {Fermion.altRightMetric | α β}ᵀ.tensor =
+    (TensorTree.add (tensorNode (basisVector ![Color.downR, Color.downR] (fun | 0 => 0 | 1 => 1))) <|
+    (smul (-1) (tensorNode (basisVector ![Color.downR, Color.downR] (fun | 0 => 1 | 1 => 0))))).tensor :=
+  altRightMetric_expand
+
 /-- The expansion of the Pauli matrices `σ^μ^a^{dot a}` in terms of basis vectors. -/
 lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
     basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0)
@@ -312,6 +384,27 @@ lemma pauliMatrix_basis_expand : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
     | (1 : Fin 3) => rfl
     | (2 : Fin 3) => rfl
 
+lemma pauliMatrix_basis_expand_tree : {PauliMatrix.asConsTensor | μ α β}ᵀ.tensor =
+   (TensorTree.add (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 0 | 2 => 0))) <|
+    TensorTree.add (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 0 | 1 => 1 | 2 => 1))) <|
+    TensorTree.add (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 0 | 2 => 1))) <|
+   TensorTree.add (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 1 | 1 => 1 | 2 => 0))) <|
+    TensorTree.add (smul (-I) (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 0 | 2 => 1)))) <|
+    TensorTree.add (smul I (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 2 | 1 => 1 | 2 => 0)))) <|
+    TensorTree.add (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 0 | 2 => 0))) <|
+    (smul (-1) (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] (fun | 0 => 3 | 1 => 1 | 2 => 1))))).tensor := by
+  rw [pauliMatrix_basis_expand]
+  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, add_tensor, tensorNode_tensor,
+    smul_tensor, neg_smul, one_smul]
+  rfl
 
 end complexLorentzTensor
 end Fermion
diff --git a/HepLean/Tensors/ComplexLorentz/Lemmas.lean b/HepLean/Tensors/ComplexLorentz/Lemmas.lean
index 0920f73..ad3d7aa 100644
--- a/HepLean/Tensors/ComplexLorentz/Lemmas.lean
+++ b/HepLean/Tensors/ComplexLorentz/Lemmas.lean
@@ -12,6 +12,8 @@ import HepLean.Tensors.Tree.NodeIdentities.PermContr
 import HepLean.Tensors.Tree.NodeIdentities.ProdComm
 import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
 import HepLean.Tensors.Tree.NodeIdentities.ContrContr
+import HepLean.Tensors.ComplexLorentz.Basis
+import LLMLean
 /-!
 
 ## Lemmas related to complex Lorentz tensors.
@@ -32,86 +34,6 @@ noncomputable section
 
 namespace Fermion
 
-/-- The vectors forming a basis of
-  `complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down])`.
-  Not proved it is a basis yet. -/
-def coCoBasis (b : Fin 4 × Fin 4) :
-    complexLorentzTensor.F.obj (OverColor.mk ![Color.down, Color.down]) :=
-  PiTensorProduct.tprod ℂ (fun i => Fin.cases (Lorentz.complexCoBasisFin4 b.1)
-  (fun i => Fin.cases (Lorentz.complexCoBasisFin4 b.2) (fun i => i.elim0) i) i)
-
-lemma coCoBasis_eval (e1 e2 : Fin (complexLorentzTensor.repDim Color.down)) (i : Fin 4 × Fin 4) :
-    complexLorentzTensor.castFin0ToField
-      ((complexLorentzTensor.evalMap 0 e2) ((complexLorentzTensor.evalMap 0 e1) (coCoBasis i))) =
-    if i = (e1, e2) then 1 else 0 := by
-  simp only [coCoBasis]
-  have h1 := @TensorSpecies.evalMap_tprod complexLorentzTensor _ (![Color.down, Color.down]) 0 e1
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Functor.id_obj,
-      OverColor.mk_hom, Function.comp_apply, cons_val_zero, Fin.cases_zero, Fin.cases_succ] at h1
-  erw [h1]
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Functor.id_obj, OverColor.mk_hom,
-    Fin.cases_zero, Fin.cases_succ, _root_.map_smul, smul_eq_mul]
-  erw [TensorSpecies.evalMap_tprod]
-  simp only [Fin.isValue, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.succAbove_zero, Functor.id_obj,
-    OverColor.mk_hom, Function.comp_apply, Fin.succ_zero_eq_one, cons_val_one, head_cons,
-    Fin.cases_zero, Fin.zero_succAbove, Fin.cases_succ, _root_.map_smul, smul_eq_mul]
-  erw [complexLorentzTensor.castFin0ToField_tprod]
-  simp only [Fin.isValue, mul_one]
-  change (Lorentz.complexCoBasisFin4.repr (Lorentz.complexCoBasisFin4 i.1)) e1 *
-    (Lorentz.complexCoBasisFin4.repr (Lorentz.complexCoBasisFin4 i.2)) e2 = _
-  simp only [Basis.repr_self]
-  rw [Finsupp.single_apply, Finsupp.single_apply]
-  rw [@ite_zero_mul_ite_zero]
-  simp only [mul_one]
-  congr
-  simp_all only [Fin.isValue, Fin.succAbove_zero, Fin.zero_succAbove, eq_iff_iff]
-  obtain ⟨fst, snd⟩ := i
-  simp_all only [Fin.isValue, Prod.mk.injEq]
-
-lemma coMetric_expand : {Lorentz.coMetric | μ ν}ᵀ.tensor =
-    coCoBasis (0, 0) - coCoBasis (1, 1) - coCoBasis (2, 2) - coCoBasis (3, 3) := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, constTwoNode_tensor,
-    Action.instMonoidalCategory_tensorObj_V, Action.instMonoidalCategory_tensorUnit_V,
-    Functor.id_obj, Fin.isValue]
-  erw [Lorentz.coMetric_apply_one, Lorentz.coMetricVal_expand_tmul]
-  simp only [Fin.isValue, map_sub]
-  congr 1
-  congr 1
-  congr 1
-  all_goals
-    erw [pairIsoSep_tmul, coCoBasis]
-    simp only [Nat.reduceAdd, Nat.succ_eq_add_one, OverColor.mk_hom, Functor.id_obj, Fin.isValue,
-      Lorentz.complexCoBasisFin4, Basis.coe_reindex, Function.comp_apply]
-    rfl
-
-/-- The covariant Lorentz metric is symmetric. -/
-lemma coMetric_symm : {Lorentz.coMetric | μ ν = Lorentz.coMetric | ν μ}ᵀ := by
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, perm_tensor]
-  rw [coMetric_expand]
-  simp only [TensorSpecies.F, Nat.succ_eq_add_one, Nat.reduceAdd, Functor.id_obj, Fin.isValue,
-    map_sub]
-  simp only [coCoBasis, Nat.succ_eq_add_one, Nat.reduceAdd, Functor.id_obj, OverColor.mk_hom,
-    Lorentz.complexCoBasisFin4, Fin.isValue, Basis.coe_reindex, Function.comp_apply]
-  congr 1
-  congr 1
-  congr 1
-  all_goals
-    erw [OverColor.lift.map_tprod]
-    congr 1
-    funext i
-    match i with
-    | (0 : Fin 2) => rfl
-    | (1 : Fin 2) => rfl
-
-lemma coMetric_0_0_field : {Lorentz.coMetric | 0 0}ᵀ.field = 1 := by
-  rw [field, eval_tensor, eval_tensor, coMetric_expand]
-  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue,
-    Function.comp_apply, Fin.succ_zero_eq_one, cons_val_one, head_cons, Fin.ofNat'_zero,
-    cons_val_zero, Functor.id_obj, OverColor.mk_hom, map_sub]
-  rw [coCoBasis_eval, coCoBasis_eval, coCoBasis_eval, coCoBasis_eval]
-  simp only [Fin.isValue, Prod.mk_zero_zero, ↓reduceIte, Prod.mk_one_one, one_ne_zero, sub_zero,
-    Prod.mk_eq_zero, Fin.reduceEq, and_self]
-
 set_option maxRecDepth 20000 in
 lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
     {T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
@@ -188,6 +110,785 @@ lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
   apply TensorTree.add_tensor_eq_snd
   rw [neg_tensor_eq hA, neg_tensor_eq (neg_perm _ _), neg_neg]
 
+/-!
+
+## The contraction of Pauli matrices with Pauli matrices
+
+And related results.
+
+-/
+open complexLorentzTensor
+
+def leftMetricMulRightMap := (Sum.elim ![Color.upL, Color.upL]  ![Color.upR, Color.upR]) ∘ finSumFinEquiv.symm
+
+lemma leftMetric_mul_rightMetric : {Fermion.leftMetric | α α' ⊗ Fermion.rightMetric | β β'}ᵀ.tensor
+    = basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    - basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    - basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [prod_tensor_eq_fst (leftMetric_expand_tree)]
+  rw [prod_tensor_eq_snd (rightMetric_expand_tree)]
+  rw [prod_add_both]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_prod _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_tensor_eq <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| smul_eq_one _ _ (by simp)]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_prod _ _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_fst <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_fst <| add_tensor_eq_snd <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| prod_basisVector_tree _ _]
+  rw [← add_assoc]
+  simp only [add_tensor, smul_tensor, tensorNode_tensor]
+  change _ =  basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    +- basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector leftMetricMulRightMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)
+  congr 1
+  congr 1
+  congr 1
+  all_goals
+    congr
+    funext x
+    fin_cases x <;> rfl
+
+
+def pauliMatrixLowerMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
+    Fin.succAbove 0 ∘ Fin.succAbove 1)
+
+abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) := (Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
+
+lemma pauliMatrix_contr_expand {n : ℕ} {c : Fin n → complexLorentzTensor.C}
+    (t : TensorTree complexLorentzTensor c) (i : Fin (n + 3)) (j : Fin (n +2))
+    (h : (pauliMatrixContrMap c) (i.succAbove j) = complexLorentzTensor.τ ((pauliMatrixContrMap c) i)) :
+    (contr i j h (TensorTree.prod t (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+      PauliMatrix.asConsTensor))).tensor = (
+    (contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 0 | 2 => 0)))).add
+    ((contr i j h (t.prod (tensorNode
+       (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 0 | 1 => 1 | 2 => 1)))).add
+    ((contr i j h (t.prod (tensorNode
+       (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1  | 1 => 0 | 2 => 1)))).add
+    ((contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 1 | 1 => 1 | 2 => 0)))).add
+   ((TensorTree.smul (-I) (contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 0 | 2 => 1))))).add
+    ((TensorTree.smul I (contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 2 | 1 => 1 | 2 => 0))))).add
+    ((contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 0 | 2 => 0)))).add
+    (TensorTree.smul (-1) (contr i j h (t.prod (tensorNode
+      (basisVector ![Color.up, Color.upL, Color.upR] fun | 0 => 3 | 1 => 1 | 2 => 1)))))))))))).tensor := by
+  rw [contr_tensor_eq <| prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
+  rw [contr_tensor_eq <| prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <|  prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| prod_add _ _ _]
+  rw [contr_tensor_eq <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd  <| prod_add _ _ _]
+  /- Moving smuls. -/
+  rw [contr_tensor_eq  <| add_tensor_eq_snd  <| add_tensor_eq_snd  <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
+  rw [contr_tensor_eq  <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd  <| add_tensor_eq_snd <| add_tensor_eq_fst <| prod_smul _ _ _]
+  rw [contr_tensor_eq  <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd  <| add_tensor_eq_snd<| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| prod_smul _ _ _]
+  /- Moving contr over add. -/
+  rw [contr_add]
+  rw [add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| contr_add _ _]
+  /- Moving contr over smul. -/
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <|  add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <|  add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_smul _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+    add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd  <| add_tensor_eq_snd <|
+    contr_smul _ _]
+  rfl
+
+lemma pauliMatrix_contr_down_0 :
+    (contr 0 1 rfl (((tensorNode (basisVector ![Color.down, Color.down] fun x => 0)).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+      PauliMatrix.asConsTensor)))).tensor
+    = basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_down_1 : (contr 0 1 rfl
+    (((tensorNode (basisVector ![Color.down, Color.down] fun x => 1)).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+      PauliMatrix.asConsTensor)))).tensor
+    = basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_pos _ rfl]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ rfl]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_down_2 : (contr 0 1 rfl
+    (((tensorNode (basisVector ![Color.down, Color.down] fun x => 2)).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+      PauliMatrix.asConsTensor)))).tensor
+    = (- I) • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
+    + (I) •  basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_down_3 : (contr 0 1 rfl
+    (((tensorNode (basisVector ![Color.down, Color.down] fun x => 3)).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+      PauliMatrix.asConsTensor)))).tensor
+    =  basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
+    + (- 1 : ℂ) • basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_pos _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+def pauliMatrixContrPauliMatrixMap := ((Sum.elim
+        ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm) ∘
+          Fin.succAbove 0 ∘ Fin.succAbove 1)
+        ![Color.up, Color.upL, Color.upR] ∘
+      ⇑finSumFinEquiv.symm) ∘
+    Fin.succAbove 0 ∘ Fin.succAbove 2)
+
+lemma pauliMatrix_contr_lower_0_0_0 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_0_1_1 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+      <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+
+lemma pauliMatrix_contr_lower_1_0_1 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_pos _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_1_1_0 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_pos _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+  · congr 1
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_2_0_1 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor =
+      (-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    + (I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+lemma pauliMatrix_contr_lower_2_1_0 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor =
+      (-I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + (I) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_neg _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+
+lemma pauliMatrix_contr_lower_3_0_0 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor =
+       basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 0 | 3 => 0)
+    + (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 0 | 2 => 1 | 3 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_pos _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+
+lemma pauliMatrix_contr_lower_3_1_1 :  (contr 0 2 rfl
+    (((tensorNode (basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1))).prod
+    (constThreeNodeE complexLorentzTensor Color.up Color.upL Color.upR
+    PauliMatrix.asConsTensor)))).tensor =
+       basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 0 | 3 => 0)
+    + (-1 : ℂ) • basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 1 | 2 => 1 | 3 => 1) := by
+  rw [pauliMatrix_contr_expand]
+  /- Product of basis vectors . -/
+  rw [add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_tensor_eq
+    <| prod_basisVector_tree _ _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|  add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| smul_tensor_eq
+    <| contr_tensor_eq <| prod_basisVector_tree _ _]
+  /- Contracting basis vectors. -/
+  rw [add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst
+    <| contr_basisVector_tree_neg _ ]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_fst <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <|  add_tensor_eq_fst <| smul_tensor_eq <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <|add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| smul_tensor_eq
+    <| contr_basisVector_tree_neg _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd
+    <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_fst <| contr_basisVector_tree_pos _]
+  rw [add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <| add_tensor_eq_snd <|
+     smul_tensor_eq <| contr_basisVector_tree_pos _]
+  /- Simplifying. -/
+  simp only [smul_tensor, add_tensor, tensorNode_tensor]
+  simp only [smul_zero, add_zero, zero_add]
+  congr 1
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+  · congr 2
+    funext k
+    fin_cases k <;> rfl
+
+
+
+/-
+lemma pauliMatrix_lower :
+    {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β}ᵀ.tensor
+    = basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 0 | 2 => 0)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 0 | 1 => 1 | 2 => 1)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 0 | 2 => 1)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 1 | 1 => 1 | 2 => 0)
+    - I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 0 | 2 => 1)
+    + I • basisVector pauliMatrixLowerMap (fun | 0 => 2 | 1 => 1 | 2 => 0)
+    + basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 0 | 2 => 0)
+    - basisVector pauliMatrixLowerMap (fun | 0 => 3 | 1 => 1 | 2 => 1) := by
+  rw [contr_tensor_eq <| prod_tensor_eq_fst <| coMetric_basis_expand_tree]
+  rw [contr_tensor_eq <| prod_tensor_eq_snd <| pauliMatrix_basis_expand_tree]
+
+  sorry -/
+
+
+lemma pauliMatrix_contract_pauliMatrix :
+  {Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | μ α β ⊗ PauliMatrix.asConsTensor | ν α' β'}ᵀ.tensor
+  = basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 0 | 3 => 1)
+    - basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 0 | 1 => 1 | 2 => 1 | 3 => 0)
+    - basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 0 | 3 => 1)
+    + basisVector pauliMatrixContrPauliMatrixMap (fun | 0 => 1 | 1 => 0 | 2 => 1 | 3 => 0)  := by
+
 end Fermion
 
 end
diff --git a/HepLean/Tensors/Tree/Basic.lean b/HepLean/Tensors/Tree/Basic.lean
index 6ce4808..6c5c505 100644
--- a/HepLean/Tensors/Tree/Basic.lean
+++ b/HepLean/Tensors/Tree/Basic.lean
@@ -679,6 +679,8 @@ lemma neg_tensor (t : TensorTree S c) : (neg t).tensor = - t.tensor := rfl
 lemma eval_tensor {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : ℕ) (t : TensorTree S c) :
     (eval i e t).tensor = (S.evalMap i (Fin.ofNat' e Fin.size_pos')) t.tensor := rfl
 
+lemma smul_tensor {c : Fin n → S.C} (a : S.k) (T : TensorTree S c) :
+    (smul a  T).tensor = a • T.tensor:= rfl
 /-!
 
 ## Equality of tensors and rewrites.
@@ -737,6 +739,11 @@ lemma neg_tensor_eq {T1 T2 : TensorTree S c} (h : T1.tensor = T2.tensor) :
   simp only [neg_tensor]
   rw [h]
 
+lemma smul_tensor_eq {T1 T2 : TensorTree S c} {a : S.k} (h : T1.tensor = T2.tensor) :
+    (smul a T1).tensor = (smul a T2).tensor := by
+  simp only [smul_tensor]
+  rw [h]
+
 /-- A structure containing a pair of indices (i, j) to be contracted in a tensor.
   This is used in some proofs of node identities for tensor trees. -/
 structure ContrPair {n : ℕ} (c : Fin n.succ.succ → S.C) where
diff --git a/HepLean/Tensors/Tree/NodeIdentities/Basic.lean b/HepLean/Tensors/Tree/NodeIdentities/Basic.lean
index be244fb..a047b4c 100644
--- a/HepLean/Tensors/Tree/NodeIdentities/Basic.lean
+++ b/HepLean/Tensors/Tree/NodeIdentities/Basic.lean
@@ -117,12 +117,26 @@ lemma perm_eq_of_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
 
 /-!
 
-## Additive identities
+## Vector based identities
 
 These identities are related to the fact that all the maps are linear.
 
 -/
 
+lemma smul_smul (t : TensorTree S c) (a b : S.k) :
+    (smul a (smul b t)).tensor = (smul (a * b) t).tensor := by
+  simp [smul_tensor]
+  exact _root_.smul_smul a b t.tensor
+
+lemma smul_one (t : TensorTree S c) :
+    (smul 1 t).tensor = t.tensor := by
+  simp [smul_tensor]
+
+lemma smul_eq_one (t : TensorTree S c) (a : S.k) (h : a = 1) :
+    (smul a t).tensor = t.tensor := by
+  rw [h]
+  exact smul_one _
+
 /-- The addition node is commutative. -/
 lemma add_comm (t1 t2 : TensorTree S c) : (add t1 t2).tensor = (add t2 t1).tensor := by
   simp only [add_tensor]
@@ -147,4 +161,48 @@ lemma add_eval {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : ℕ) (t
     (add (eval i e t) (eval i e t1)).tensor = (eval i e (add t t1)).tensor := by
   simp only [add_tensor, eval_tensor, Nat.succ_eq_add_one, map_add]
 
+lemma contr_add {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
+    {h : c (i.succAbove j) = S.τ (c i)} (t1 t2 : TensorTree S c) :
+  (contr i j h (add t1 t2)).tensor = (add (contr i j h t1) (contr i j h t2)).tensor := by
+  simp [contr_tensor, add_tensor]
+
+lemma contr_smul {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
+    {h : c (i.succAbove j) = S.τ (c i)} (a : S.k) (t : TensorTree S c) :
+    (contr i j h (smul a t)).tensor = (smul a (contr i j h t)).tensor := by
+  simp [contr_tensor, smul_tensor]
+
+@[simp]
+lemma add_prod {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+    (t1 t2 : TensorTree S c) (t3 : TensorTree S c1) :
+    (prod (add t1 t2) t3).tensor = (add (prod t1 t3) (prod t2 t3)).tensor := by
+  simp only [prod_tensor, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
+    Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
+    Action.FunctorCategoryEquivalence.functor_obj_obj, add_tensor, add_tmul, map_add]
+
+@[simp]
+lemma prod_add {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+    (t1 : TensorTree S c) (t2 t3 : TensorTree S c1) :
+    (prod t1 (add t2 t3)).tensor = (add (prod t1 t2) (prod t1 t3)).tensor := by
+  simp only [prod_tensor, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
+    Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
+    Action.FunctorCategoryEquivalence.functor_obj_obj, add_tensor, tmul_add, map_add]
+
+lemma smul_prod {n m: ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+    (a : S.k) (t1 : TensorTree S c) (t2 : TensorTree S c1) :
+    ((prod (smul a t1) t2)).tensor = (smul a (prod t1 t2)).tensor := by
+  simp [prod_tensor, smul_tensor, tmul_smul, smul_tmul, map_smul]
+
+lemma prod_smul {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+  (a : S.k) (t1 : TensorTree S c) (t2 : TensorTree S c1) :
+    (prod t1 (smul a t2)).tensor = (smul a (prod t1 t2)).tensor := by
+  simp [prod_tensor, smul_tensor, tmul_smul, smul_tmul, map_smul]
+
+lemma prod_add_both {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+    (t1 t2 : TensorTree S c) (t3 t4 : TensorTree S c1) :
+    (prod (add t1 t2) (add t3 t4)).tensor = (add (add (prod t1 t3) (prod t1 t4))
+    (add (prod t2 t3) (prod t2 t4))).tensor := by
+  rw [add_prod]
+  rw [add_tensor_eq_fst (prod_add _ _ _)]
+  rw [add_tensor_eq_snd (prod_add _ _ _)]
+
 end TensorTree