The quaternary system comprising sodium dodecyl sulfate/ water/pentanol/dodecane has been studied at high oil content and at two different water-to-surfactant ratios. Of particular interest is a highly flow birefringent, isotropic fluid phase, which occupies an island in the pseudo-ternary phase diagrams between the L2 phase and the L3 phase and which is in close proximity to the dilute lamellar phase. In order to investigate its microstructure, the self-diffusion coefficient of the solvent was measured in this phase using Fourier transform pulsed-field-gradient spin-echo NMR spectroscopy, and its value was found to be significantly lower than those in the L3 phase of the same system. Comparative conductivity measurements were also performed, and they showed that the conducting material in the phase forms an infinitely connected network. In conclusion, it seems reasonable to describe the phase microstructure as a connected perforated bilayer or possibly a connected three-dimensional network of reverse cylinders. The conductivity of the Lx phase decreases strongly upon stirring. Taking into account the flow birefringence, this is interpreted as a shear-induced phase transition to a lamellar phase.