 Segre description of a smooth quadric surface

Over an algebraically closed field, a smooth four-variable projective quadratic equation can be written $x_0{x}_3-x_1{x}_2=0$, including in characteristic two. Its quadratic form splits into two hyperbolic planes: choose an isotropic vector and its paired vector, adjust the latter to be isotropic, and repeat on the orthogonal complement. In characteristic two, smoothness forces the alternating polar form in dimension four to have full rank, since a positive-dimensional radical has a nonzero isotropic vector over the algebraically closed field and would give a singular point. The Segre embedding $([s_0:s_1],[t_0:t_1])\mapsto [s_0{t}_0:s_0{t}_1:s_1{t}_0:s_1{t}_1]$ then gives the isomorphism. Each matrix of coordinates has rank one, which supplies the inverse pair of projective factors.