When does the associated graded Lie algebra of an arrangement group decompose?

Let \mathcal{A} be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \mathfrak{H} . Suppose \mathfrak{H}_3 is a free abelian group of minimum possible rank, given the values the Möbius function \mu\colon \mathcal{L}_2\to \mathbb{Z} takes on the rank 2 flats of \mathcal{A} . Then the associated graded Lie algebra of G decomposes (in degrees \ge 2 ) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by \phi_r(G)=\sum _{X\in \mathcal{L}_2} \phi_r(F_{\mu(X)}) , for r\ge 2 . We illustrate this new Lower Central Series formula with several families of examples.