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.

  1. What are possible orders of $x$ when we know $a,b$ ?
  2. 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.
  3. Classify possible 2-dimensional subalgebras (with $1$) generated by $x$.
  4. 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 $Q_{1,-1}$.

  1. The question is how could we find other similar nice formulas for conjugacy classes in $G_2(q)$ ?
  2. 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 $F_{q^2}$ ?

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