Swollen cubic lyotropic ternary phases with Pn3m symmetry and reduced hardness were obtained from a specific binary mixture of cubic phase-forming (phytantriol) and lamellar phase-forming (decaglycerol monooleate) compounds. The microstructures were determined by using a small-angle X-ray scattering technique. The softness and temperature-induced phase transitions were investigated by means of rheology. The incorporation of a surface-active fragrance compound (linalool) at concentrations up to 6%wt induced a structural transition toward a softer Im3m bulk cubic phase with longer water channels. Higher linalool concentrations allowed the spontaneous dispersion of the bulk cubic phase into microscopic particles with a cubic structure (cubosomes).