AbstractLetXbe a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the$S^1$-equivariant homology$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $of the free loop space ofXpreserves the Hodge decomposition of$ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $, making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].