A search is conducted for a beyond-the-Standard-Model boson using events where a Higgs boson with mass 125 GeV decays to four leptons (l = e or mu). This decay is presumed to occur via an intermediate state which contains one or two on-shell, promptly decaying bosons: H -> ZX/XX -> 4l , where X is a new vector boson Z(d) or pseudoscalar a with mass between 1 and 60 GeV. The search uses pp collision data collected with the ATLAS detector at the LHC with an integrated luminosity of 36.1 fb(-1) at a centre-of-mass energy root s = 13TeV. No significant excess of events above Standard Model background predictions is observed; therefore, upper limits at 95% confidence level are set on model-independent fiducial cross-sections, and on the Higgs boson decay branching ratios to vector and pseudoscalar bosons in two benchmark models.