Unit pair symmetric factorization

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.GCD
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination

section
open scoped NumberField
open Module

namespace D1Bound5


theorem _root_.D1Bound5.unit_pair_symmetric_factorization
    {R : Type*} [CommRing R] (a b x y v w : R)
    (hx : x ^ 2 = 1 - b) (hy : y ^ 2 = 1 + b)
    (hv : v ^ 2 = 1 - a * b) (hw : w ^ 2 = 1 + a * b) :
    ((2 * a ^ 2 * b ^ 2 - a ^ 2 - 1) - 2 * a * x * y * v * w) *
      ((2 * a ^ 2 * b ^ 2 - a ^ 2 - 1) + 2 * a * x * y * v * w) =
        (a ^ 2 - 1) ^ 2 := by sorry









end D1Bound5
end