The structural and the electronic properties of a hypothetical three-dimensional (3D) all- s p2 carbon phase, called K4, are investigated using first-principles calculations. The cohesive energy per atom for this structure is found to be 1.3 eV lower than for graphite and diamond, but also more than 0.3 eV lower than for the previously proposed H-6, bct-4, and C-20 phases, also 3D all- s p2 carbon forms. The calculated bulk modulus for the K4 crystal is slightly higher than for the C-20 structure but lower than for the H-6 and bct-4 solids. Our analysis reveals that the relatively low cohesive energy and bulk modulus of the K4 phase are due to the disruption of the π bonding states. The calculated density of states and band structure show that the K4 phase is metallic.