  Jet feedback from active galactic nuclei (AGN) harboured by brightest cluster
galaxies is expected to play a fundamental role in regulating cooling in the
intracluster medium (ICM). While observations and theory suggest energy within
jet lobes balances ICM radiative losses, the modus operandi of energy
communication with the ICM remains unclear. We present simulations of very high
resolution AGN-driven jets launching in a live, cosmological galaxy cluster,
within the moving mesh code AREPO. As the jet propagates through the ICM the
majority of its energy, which is initially in the kinetic form, thermalizes
quickly through internal shocks and inflates lobes of very hot gas. The jets
effectively heat the cluster core, with $PdV$ work and weather-aided mixing
being the main channels of energy transfer from the lobes to the ICM, while
strong shocks and turbulence are subdominant. We additionally present detailed
mock X-ray maps at different stages of evolution, revealing clear cavities
surrounded by X-ray bright rims, with lobes being detectable for up to $\sim
10^8$ yr even when magnetic draping is ineffective. We find bulk motions in the
cluster can significantly affect lobe propagation, offsetting them from the jet
direction and imparting bulk velocities that can dominate over the buoyantly
rising motion.
