As I understand, your question is about relations on the generators of the Garland lattice for the hyperloop algebra of $\mathfrak g$. Maybe the following papers will help you:
H.Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480--551.
D. Jakelic, A. Moura, Finite-dimensional representations of hyper loop algebras, Pacific J. Math. 233 (2007), 371--402.