A search for CP violation in the decay kinematics and vector-boson fusion production of the Higgs boson is performed in the H → ZZ$^{*}$ → 4ℓ (ℓ = e, μ) decay channel. The results are based on proton-proton collision data produced at the LHC at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector from 2015 to 2018, corresponding to an integrated luminosity of 139 fb$^{−1}$. Matrix element-based optimal observables are used to constrain CP-odd couplings beyond the Standard Model in the framework of Standard Model effective field theory expressed in the Warsaw and Higgs bases. Differential fiducial cross-section measurements of the optimal observables are also performed, and a new fiducial cross-section measurement for vector-boson-fusion production is provided. All measurements are in agreement with the Standard Model prediction of a CP-even Higgs boson.[graphic not available: see fulltext]