Invariants of cubic forms in four variables

We give a new proof of the completeness of the classical system of invariants of cubic forms in four variables. We prove that the orbit-space of the space of cubic forms in four variables with respect to the action of $\mathrm{SL}(4)$ is a hypersurface and that the projectivization of the factor with respect to the natural action of $\mathbf{C}^*$ is a rational variety.