With the aim of developing a single-phase light-weight HEA (LWHEA), the quaternary Al-Li-Mg-Zn alloy system is investigated with the initial selection of elements based on typical HEA formation guidelines. Beginning with an equiatomic four-component composition, AlLiMgZn, and systematically altering elemental concentrations revealed that the Mg-rich region of this quaternary space would be most promising for producing a single-phase solid solution LWHEA. Alloy compositions were refined by replicating the compositions of individual phases within the as-cast microstructure in a series of Mg-rich quaternary alloys, thereby effectively limiting the number of iterative steps and associated processing required to reach a single-phase alloy composition. Three crystal structures were identified in different compositional regions, consisting of hexagonal, monoclinic, and icosahedral quasicrystal. The underlying origin of the resultant multicomponent crystal structures and structural ordering are explored, showing a strong correlation to the electronic structure in the formation of the quasicrystal.