A search for exotic decays of the Higgs boson into a pair of low-mass scalars that subsequently decay into tau -leptons, H -> aa -> tau(+)tau(-)tau(+)tau(-), is presented. In models with Yukawa-like couplings, the decay to tau-leptons is favoured for light a-bosons, with mass in the range of 2m(tau) < m(a) < 2m(b). Results are presented in the range of 4 GeV < m(a) < 15 GeV using the 140 fb(-1) of proton-proton collisions at root s = 13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider. This search focuses on the scenario where, for both di-tau pairs, one of the tau-leptons decays to hadrons and neutrinos, while the other decays to a muon and neutrinos. In this mass range, the a -> tau(+)tau(-) is Lorentz-boosted and a dedicated muon removal technique is used to reconstruct the di-tau pairs. No significant excess above the Standard Model background prediction is observed. Upper limits on (sigma(H)/sigma(SM)(H)) x B(H -> aa -> 4 tau) at 95% confidence level are provided, ranging from 0.03 to 0.10 depending on the a-boson mass.