The $SO(3,1)$-orbits in the light cone of the 2-fold exterior power of the Minkowski 4-space

Two special neutral hypersurfaces ${\mathcal{L}}_{\pm }$ in the light cone $L({\bigwedge }^2{E}_1^4)$ studied in [1], [3] are $SO(3,1)$-orbits. In this paper, we see that each $SO(3,1)$-orbit in $L({\bigwedge }^2{E}_1^4)$ is either a neutral hypersurface homothetic to one of ${\mathcal{L}}_{\pm }$ in $L({\bigwedge }^2{E}_1^4)$ or a hypersurface with a two-dimensional involutive distribution where the induced metric is degenerate. The difference between these hypersurfaces can be understood in terms of the stabilizer and the $r$-slice of $L({\bigwedge }^2{E}_1^4)$ for $r>0$.