Let be a complex hyperplane arrangement, with fundamental group and holonomy Lie algebra . Suppose is a free abelian group of minimum possible rank, given the values the Möbius function takes on the rank flats of . Then the associated graded Lie algebra of decomposes (in degrees ) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by , for . We illustrate this new Lower Central Series formula with several families of examples.