Once octonions product is studied, together with the relations with $Spin(8)$ and $SO(8)$ geometry (see for instance Robert Bryant's notes), one realises that the key fact bringing all the phenomena of triality etc. is the following: that there exist some octonions $u_1,\ldots,u_k$ such that

$$R_{u_k}\ldots R_{u_1}=-Id\qquad\qquad L_{u_k}\ldots L_{u_1}=Id,$$

where $L_u$ and $R_u$ denote left and right product with $u$ respectively.

This may be found by the Lie group properties of $Spin(8)$ and $SO(8)$. But

**May some set $u_1,\ldots,u_k$ be described explicitely?**

Any idea is welcome.

**EDIT:**

Although it is not directly related to the question, it is interesting to note the vital importance of the claim. Following Briant's notes, we consider the group of maps $\mathbb{O}\oplus\mathbb{O}\longrightarrow\mathbb{O}\oplus\mathbb{O}$ generated by the elements $L_u\oplus R_u:(a,b)\longmapsto(ua,bu)$ with $\|u\|=1$. Of course, we are dealing with some group $G\subset SO(8)\times SO(8)$. Then: if we read the previous paragraphs, it happens that $G$ is the group $Spin(8)$ defined in terms of Clifford algebras, and the key property of this $Spin(8)$ being really a double cover of $SO(8)$ is supported partially by the fact that *the inclussion $G\subset SO(8)\times SO(8)$ is not a vacuous one, as could be the diagonal $SO(8)\subset SO(8)\times SO(8)$, because $(Id,-Id)\in G$*. That is why the sign is crucial.

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