Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN\,$\times$\,quadrupole cross terms and of the quadrupole-squared term

We derive the secular Hamiltonian of a hierarchical triple from the 1PN ADM three-body Hamiltonian by a two-stage Lie-Hori-Deprit transformation carried out entirely in canonical variables, and use it to settle the relativistic cross terms left open in Paper I. A bookkeeping rule enumerates all cont