An uniﬁed structural model has been devised  that describes both icosahedral phasesi-AlPdMn and i-AlCuFe. The model suggests that possible icosahedral structures could be found inalloys with general composition Al61:8(PdjCu)21:35(MnjFe)8:29(AljFejMn)4:28(AljCu)4:28. Applied to the systems AlCuMnFe  and AlPdFe , thisformula leads to the compositions Al66:08Cu21:35Mn8:29Fe4:28 and Al70:36Pd21:35Fe8:29 that have bothbeen synthetized by rapid quenching and annealing.The as-quenched quaternary alloy Al66:08Cu21:35Mn8:29Fe4:28 exhibits a F-icosahedral phase that trans-forms into a decagonal phase after annealing. Its 10-fold axis aligns along one of the 5-fold axes ofthe icosahedral phase. This implies a group-subgroup transformation, with the kernel group ̅5m, anal-ogous to the one observed during CFC to HCP transformations, with kernel ̅3m.The as-quenched and annealed ternary alloy Al70:36Pd21:35Fe8:29 is similar to i-AlPdMn with an iso-morphic substitution of Mn with Fe. A study of the phase diagram of the system AlPdFe  showsthat the results are compatible with the structural model proposed here in strong relations with thosepresented in  for i-AlPdMn.