Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over $\mathbb{Z}$

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 $R=\mathbb{Z}$, we prove that this map extends to a group homomorphism $\rho_{\lambda,\Z}: G(\Z) \to G^{\lambda}(\Z).$Weprovethatthekernel$K^{\lambda}$ofthemap$\rho_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$liesin$H(\C)$ and if the group homomorphism $\phi :G(\mathbb{Z})\to G(\text{C})$ is injective, then $K^{\lambda}\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.