We present a systematic study of the sensitivity of quasielastic neutrino-nucleus cross sections at intermediate energies to the strange quark sea of the nucleon. To this end, we investigate the impact of the weak strangeness form factors on the ratio of proton-to-neutron knockout, the ratio of neutral-to-charged current cross sections, on the Paschos-Wolfenstein relation, and on the longitudinal helicity asymmetry. The influence of axial as well as vector strangeness effects is discussed. For the latter, we introduce strangeness parameters from various hadron models and from a recent fit to data from parity violating electron scattering. In our model, the nuclear target is described in terms of a relativistic mean-field approach. The effects of final-state interactions on the outgoing nucleon are quantified within a relativistic multiple-scattering Glauber approach. Our results are illustrated with cross sections for the scattering of 1 GeV neutrinos and antineutrinos off a $^{12}$C target. Folding with a proposed FINeSSE (anti)neutrino energy-distribution has no qualitative influence on the overall sensitivity of the cross-section ratios to strangeness mechanisms. We show that vector strangeness effects are large and strongly $Q^2$ dependent.
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