  We present the calculation of next-to-leading-order electroweak corrections
to W-boson pair production at the LHC, taking off-shell effects of the W bosons
and their leptonic decays into account in the framework of the so-called
double-pole approximation. In detail, the lowest-order cross section and the
photonic bremsstrahlung are based on full matrix elements with four-fermion
final states, but the virtual one-loop corrections are approximated by the
leading contributions of a systematic expansion about the resonance poles of
the two W bosons. This expansion classifies the virtual corrections into
factorizable and non-factorizable corrections, the calculation of which is
described in detail. Corrections induced by photons in the initial state, i.e.
photon-photon and quark-photon collision channels, are included and based on
complete matrix elements as well. Our numerical results, which are presented
for realistic acceptance cuts applied to the W-boson decay products,
qualitatively confirm recent results obtained for on-shell W bosons and reveal
electroweak corrections of the size of tens of percent in the TeV range of
transverse momenta and invariant masses. In general, photon-photon and
quark-photon induced contributions amount to 5-10% of the full differential
result. Compared to previous predictions based on stable W bosons electroweak
corrections, however, can change by several percent because of realistic cuts
on the W-boson decay products and corrections to the decays.
