From an octonion algebra $𝕆$ over a field $k$ of characteristic not two or three, we show that the fundamental representation $\mathrm{\text{Im}}(𝕆)$ of the derivation algebra $\mathrm{\text{Der}}(𝕆)$ and the spinor representation $𝕆$ of $𝔰𝔬(\mathrm{\text{Im}}(𝕆))$ 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.

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与异常李超代数相关的特殊正交表示的几何性质

从八元代数$𝕆$在一片田野上$k$特征不是两个或三个，我们表明基本表示$\mathrm{\text{我是}}（𝕆）$推导代数的$\mathrm{\text{德尔}}（𝕆）$和旋量表示$𝕆$的$𝔰𝔬（\mathrm{\text{我是}}（𝕆））$是特殊的正交表示。它们具有来自与二元三次的相似性的特殊几何性质，并且我们表明这些表示的协变及其马修斯恒等式与法诺平面和仿射空间相关${（{\mathbb{Z}}_2）}^3$。这也允许给出特殊的李超代数的构造。