Octonionic projective plane

Example

Let $\mathbb{O}$ be the octonions, the $8$-dimensional real division algebra that is alternative but not associative. The octonionic projective plane $\mathbb{O}P^2$, also called the Cayley plane, has the following points and lines.

1. Affine points $(x, y)$ with $x, y \in \mathbb{O}$, points $(m)$ with $m \in \mathbb{O}$, and one more point $(\infty)$.
2. Lines $[m, b] = \set{(x, y) : y = x m + b} \cup \set{(m)}$ for $m, b \in \mathbb{O}$, lines $[c] = \set{(c, y) : y \in \mathbb{O}} \cup \set{(\infty)}$ for $c \in \mathbb{O}$, and the line at infinity $\set{(m) : m \in \mathbb{O}} \cup \set{(\infty)}$.

This is a projective plane. The key check concerns two affine points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \ne x_2$. They lie on a line $[m, b]$ exactly when $(x_1 - x_2) m = y_1 - y_2$ and $b = y_1 - x_1 m$. In an alternative division algebra, $a m = c$ with $a \ne 0$ has the unique solution $m = a^{-1} c$, since $a (a^{-1} c) = c$.

It is a Moufang plane, since it is coordinatized by the alternative division ring $\mathbb{O}$. It is not a Desarguesian plane: a Desarguesian plane is coordinatized by an associative division ring, and $\mathbb{O}$ is not associative.