In joint work with Xujia Chen and Sander Kupers, we construct a bracket operation on the space of framed disk bundles of fiber dimension at least 4. Kontsevich used integrals over configuration spaces to produce graph homology classes from classes of disk bundles. We prove that our bracket operation is compatible with these Kontsevich characteristic classes via the bracket operation on graph homology. Applying our bracket to Watanabe’s bundles from Borromean surgery on trivalent graphs, we obtain new disk bundles, some of which are nontrivial.