Let ( L, ∂) be a differential graded Lie algebra over the prime field F p . There exists an isomorphism of Hopf algebras H ∗(UL)≅UE , where E is a graded Lie algebra (J. Pure. Appl. Algebra 83 (1992) 237–282). Suppose that L is q-reduced for some q⩾1. We prove a generalization of a classical theorem of Sullivan (Inst. Hautes Études Sci. Publ. Math. (47) (1977) 269–331), which we use to show that there is an isomorphism of graded Lie algebras H( L, ∂)≅ E× K, where K is an abelian ( qp+ p−2)-reduced ideal. As a consequence, if X is a finite, q-connected, n-dimensional CW complex, and E X is its mod p homotopy Lie algebra (J. Pure. Appl. Algebra 83 (1992) 237–282), then there are isomorphisms (E X) m≅π m+1(X; F p) for m⩽min( q+2 p−3, pq−1).