A PBW basis for Lusztig's form of untwisted affine quantum groups

Let $\mathfrak{g}$ be an untwisted affine Kac-Moody algebra over the field $K \,$, and let $U_q(\mathfrak{g})$ be the associated quantum enveloping algebra; let $\mathfrak{U}_q(g)$ be the Lusztig's integer form of $U_q(\mathfrak{g}) \,$, generated by $q$-divided powers of Chevalley generators over a suitable subring $R$ of $K(q) \,$. We prove a Poincar\'e-Birkhoff-Witt like theorem for $\mathfrak{U}_q(\mathfrak{g}) \,$, yielding a basis over $R$ made of ordered products of $q$-divided powers of suitable quantum root vectors.