  We continue our study and classification of structures on the Fano plane
${\cal F}$ and its dual ${\cal F}^\ast$ involved in the construction of
octonions and the Lie algebra $\mathfrak g_2 (\mathbb F)$ over a field $\mathbb
F$. These are a "composition factor": ${\cal F}\times {\cal F} \to\{-1, 1\}$,
inducing an octonion multiplication, and a function $\delta^\ast : Aut({\cal
F}) \times {\cal F}^\ast \to \{-1, 1\}$ such that $g \in Aut({\cal F})$ can be
lifted to an automorphism of the octonions iff $\delta^\ast(g, \cdot)$ is the
Radon transform of a function on ${\cal F}$. We lift the action of $Aut({\cal
F})$ on ${\cal F}$ to the action of a non-trivial eight-fold covering
$Aut({\cal F})$ on a twofold covering $\hat {\cal F}$ of ${\cal F}$ contained
in the octonions. This extends tautologically to an action on the octonions by
automorphism. Finally, we associate to incident point-line pairs a generating
set of $\mathfrak g_2 (\mathbb F)$ and express brackets in terms of the
incidence geometry of ${\cal F}$ and $\epsilon$.
