  From an octonion algebra $\mathbb{O}$ over a field $k$ of characteristic not
two or three, we show that the fundamental representation ${\rm
Im}(\mathbb{O})$ of the derivation algebra ${\rm Der}(\mathbb{O})$ and the
spinor representation $\mathbb{O}$ of $\mathfrak{so}({\rm Im}(\mathbb{O}))$ are
special orthogonal representations. They have particular geometric properties
coming from their similarities with binary cubics and we show that the
covariants of these representations and their Mathews identities are related to
the Fano plane and the affine space $(\mathbb{Z}_2)^3$. This also permits to
give constructions of exceptional Lie superalgebras.
