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 frame (II, p. 202, exerc. 10); denoting by $D_{xy}$ the right passing through two distinct points $x, y$ of $\mathbf{P}(V)$, let $e, f, g$ be the points of intersection of the rights $D_{ab}$ 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 $D_{ef}$; now prove that one has $\begin{bmatrix} b & c \\ h & g \end{bmatrix} = \{-1\}$ (“theorem of the complete quadrilateral”; reduce to the case where $D_{ad}$ is the right at infinity of an affine plane). What is the corresponding result when $K$ is of characteristic 2?*