Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

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