  Let $X$ be 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{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q})
$ of the free loop space of $X$ preserves the Hodge decomposition of
$\overline{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 proposed
in our earlier work.
