feat: Add metric invariance for Real Lorentz Tensors

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⟩