∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N}
  [inst_5 : P.IsCrystallographic] (b : P.Base) [inst_6 : Fintype ι] [inst_7 : DecidableEq ι]
  {x y : Matrix (↥b.support ⊕ ι) (↥b.support ⊕ ι) R},
  x ∈ (Submodule.span R (Set.range RootPairing.GeckConstruction.h)).carrier →
    y ∈ (Submodule.span R (Set.range RootPairing.GeckConstruction.h)).carrier →
      ⁅x, y⁆ ∈ (Submodule.span R (Set.range RootPairing.GeckConstruction.h)).carrier