  Let A be a graded-commutative, connected k-algebra generated in degree 1. The
homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the
Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its
quadratic closure, we express g_A as a semi-direct product of the
well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows
us to compute the homotopy Lie algebra associated to the cohomology ring of the
complement of a complex hyperplane arrangement, provided some combinatorial
assumptions are satisfied. As an application, we give examples of hyperplane
arrangements whose complements have the same Poincar\'e polynomial, the same
fundamental group, and the same holonomy Lie algebra, yet different homotopy
Lie algebras.
