  Let \A be a complex hyperplane arrangement, with fundamental group G and
holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum
possible rank, given the values the M\"obius function \mu: \L_2\to \Z takes on
the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes
(in degrees 2 and higher) 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 \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We
illustrate this new Lower Central Series formula with several families of
examples.
