Exercise 13, § 9

In a projective plane $\mathbf{P}(V)$ over a field K of characteristic $\neq 2$, let $a, b, c, d$ be four points forming a projective basis (Exercise 10); denoting by $D_{xy}$ the line passing through two distinct points $x, y$ of $\mathbf{P}(V)$, let $e, f, g$ be the points of intersection of the lines $D_{ab}$ and $D_{cd}$, $D_{ac}$ and $D_{bd}$, $D_{ad}$ and $D_{bc}$ respectively; let $h$ be the point of intersection of $D_{bc}$ and $D_{ef}$; show that $\begin{bmatrix} b & c \\ h & g \end{bmatrix} = \{-1\}$ ("theorem of the complete quadrilateral"); reduce it to the case where $D_{ad}$ is the line at infinity of an affine plane). What is the corresponding result when K is of characteristic 2?*