This paper presents a dedicated search for exotic decays of the Higgs boson to a pair of new spin-zero particles, H -> aa, where the particle a decays to b-quarks and has a mass in the range of 20-60 GeV. The search is performed in events where the Higgs boson is produced in associationwith aW boson, giving rise to a signature of a lepton (electron or muon), missing transverse momentum, and multiple jets from b-quark decays. The analysis is based on the full dataset of pp collisions at root s = 13 TeV recorded in 2015 by theATLAS detector at theCERNLargeHadron Collider, corresponding to an integrated luminosity of 3.2 fb(-1). No significant excess of events above the Standard Model prediction is observed, and a 95% confidence-level upper limit is derived for the product of the production cross section for pp -> WH times the branching ratio for the decay H -> aa -> 4b. The upper limit ranges from 6.2 pb for an a-boson mass m(a) = 20 GeV to 1.5 pb for m(a) = 60 GeV.