**Rearranged on 2017-05-31**

What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define.

Many of the finite groups are defined using machinery from Lie groups. For example compact group $F_4$ is defined using Albert algebra over octonions. Finite group $F_4(q)$ is defined similarly by using octonions over field $\mathbb F_q$. See for example Wilson - Albert algebras and construction of the finite simple groups $F_4(q)$, $E_6(q)$ and $^2E_6(q)$ and their generic covers.

Lie group $G_2$ is defined as group of automorphisms of octonions. Finite group $G_2(q)$ is group of automorphisms of finite octonions $\mathbb O(q)$. I observed that this group is generated by elements $(L_xR_y)^2$ where $x$, $y$ are imaginary perpendicular octonions. Such element fix subalgebra $\langle x,y\rangle$. $L_x$ is left multiplication and $R_x$ is right multiplication by octonion $x$. Octonion $x$ is *imaginary* when $x\bar x=1$ and $\bar x=-x$ and $x \neq 1$. Two imaginary octonions are *perpendicular* when they commute or anticommute (equivalently product of them is imaginary). The number of imaginary octonions is $q^6-1$ for $q$ even; $q^6+(-1)^{(q - 1)/2}q^3$ for $q$ odd. By experiment in GAP I conjecture that number of perpendicular imaginary elements to fixed imaginary $u$ is $q^5-2$ for $q$ even; $q^5+(-1)^{(q + 1)/2}q^2$ for $q$ odd. As next step we can count number of involutions equal to number of quaternion subalgebras. We need to be careful in characteristic two, what the quaternion subalgebra is. There are two involution conjugacy classes in this case. The stabilizer of involution is $D_2(q)$ in case of $q$ odd. Another step would be to find $SU_3$ equivalent there. Surprisingly there are two of them $U_3(q)$ and $SL_3(q)$.

**Addition 2017-07-31**

In compact Lie group $G_2$ each element fixes some complex subalgebra of octonions. The automorphisms fixing a given subalgebra form $SU_3$. In the finite case each element of $G_2(q)$ fixes some 2-dimensional subalgebra generated by one element. The automorphisms fixing a given 2-dimensional subalgebra generated by one element form $U_3(q)$ or $SL_3(q)$ (the details to be clarified).

**End of addition**

I found generators of $S_8(2)$ using octonions $\mathbb O(2)$. The conjugacy class of size 255 consists of elements $\{L_xR_xS\}$ for invertible $x$, $I+L_uR_u$ for $uu=0$ and $I+(S+L_\bar v)R_v$ for other zero divisor $v$ (satisfying $vv=v$) ($S$ is matrix of conjugation). Group $S_8(2)$ is equal to $O_9(2)$, so it corresponds to $Spin_9$ subgroup in $F_4$. Look at $F_4$ as automorphisms of projective plane $\mathbb OP^2$ and $Spin_9$ as automorphisms of $\mathbb OP^1$. How could we define $O_9(q) \subset F_4(q)$ using finite octonions $\mathbb O(q)$?

Similarly how to define $O_{10}^+(q) \subset E_6(q)$ and $O_{10}^-(q) \subset {^2E_6(q)}$ using finite octonions? In compact case we may look at $E_6$ as automorphisms of "projective plane $\mathbb C \otimes \mathbb OP^2$". See Baez. I put quotes, because not all mathematicians believe in this "projective plane". In compact case we have $Spin_{10}$ and $E_6$ as automorphisms of $1$- and $2$-dimensional planes over $\mathbb C \otimes \mathbb O$.

Has anybody tried to define $E_7(q)$ and $E_8(q)$ using octonions? What are the $O_{12}$ and $O_{16}$ embedded?

Leech lattice can be seen as result of 819 points on $\mathbb OP^2$. See the definition in Wilson - Octonions and the Leech lattice of the Leech lattice. It is defined as union of 819 distinct $E_8$ lattices.

I don't know how to ask about application of finite octonions to define sporadic groups. I encountered big criticism here.

imaginary perpendicularoctonions? $\endgroup$ – Stefan Kohl May 26 '17 at 19:04