After great success of this question I would like to continue this thread. Choose any element $x$ in algebra $\mathbb O_q$ of octonions over finite field $\mathbb F_q$. There is following equation satisfied $x^2=ax+b$ for certain $a,b\in\mathbb F_q$, $a$ is trace of $x$ (which is $x+\bar x)$ and $-b$ is norm of $x$ (which is $x\bar x$). Let us denote by $Q_{b,a}$ set of octonions with norm $b$ and trace $a$. We can define scalar product in octonions by following formula: $<x,y>=tr(x\bar y)$. We call two octonions perpendicular when their scalar product is zero.
Now there are following questions which one can ask or statements which we would like to prove.
- What are possible orders of $x$ when we know $a,b$ ?
- Element $x$ belong to some subalgebra $\mathbb H_q=M_2\mathbb F_q$, so possible orders of octonions are the same as possible orders of quaternions, which are defined here.
- Classify possible 2-dimensional subalgebras (with $1$) generated by $x$.
- Classify possible 3-,4-dimensional subalgebras generated by two octonions $x,y$.
In this question on mathoverflow I am describing automorphisms of order $2$ of octonions of shape $(L_aR_b)^2$ for perpendicular elements $a,b$ from $Q_{10}$. Another automorphisms of order $3$ can be defined by $L_uR_{u^{-1}}$ for $u$ in $Q_{1,-1}$.
- The question is how could we find other similar nice formulas for conjugacy classes in $G_2(q)$ ?
- Using the other question result and defining octonions by double Cayley-Dickson process starting from $F_{q^2}$ can we look at automorphisms of octonions as $4\times 4$ matrices over $F_{q^2}$ ?
