Complexification of Tetrahedron and Pseudo-Projective Transformations

It is proved that octahedron is the complex version of tetrahedron in the following sense. The symmetry group of tetrahedron, $A_3$, can be regarded as the group of projective transformations of the space $\mathbb{R}\mathbb{P}^2$ that preserve a quadruple of points. This group can be extended to the group of transformations of the space $\mathbb{РЎ}\mathbb{P}^2$ that preserve a quadruple of points and take complex lines into complex ones. This group turns out to be the symmetry group $B_3$ of octahedron.