The physics of neutrino-nucleus cross sections is a critical probe of the Standard Model and beyond. A precise understanding is also needed to accurately deduce astrophysical neutrino spectra. At energies above ∼ 5 GeV, the cross section is dominated by deep inelastic scattering, mediated by weak bosons. In addition, there are subdominant processes where the hadronic coupling is through virtual photons, γ * : (on-shell) W -boson production (e.g., where the underlying interaction is ν +γ * → − +W + ) and trident production (e.g., where it is ν +γ * → ν + − 1 + + 2 ). These processes become increasingly relevant at TeV-PeV energies. We undertake the first systematic approach to these processes (and those with hadronic couplings through virtual W and Z bosons), treating them together, avoiding common approximations, considering all neutrino flavors and final states, and covering the energy range 10 -10 8 GeV. In particular, we present the first complete calculation of W -boson production and the first calculation of trident production at TeV-PeV energies. When we use the same assumptions as in prior work, we recover all of their major results. In a companion paper [1], we show that these processes should be taken into account for IceCube-Gen2.