Uniqueness of representation-theoretic hyperbolic Kac-Moody groups over Z

For a simply laced and hyperbolic Kac-Moody group G = G(R) over a commutative ring R with 1, we consider a map from a finite presentation of G(R) obtained by Allcock and Carbone to a representation-theoretic construction G(lambda)(R) corresponding to an integrable representation V-lambda with dominant integral weight lambda. When R = Z, we prove that this map extends to a group homomorphism rho(lambda),(Z) : G(Z) -> G(lambda)(Z). We prove that the kernel K-lambda of rho(lambda), (Z) lies in H(C) and if the natural group homomorphism phi : G(Z) -> G(C) is injective, then K-lambda <= H(Z) congruent to (Z/2Z)(rank)(G).