When looking into sizes of finite simple group of "Lie type", I observed that power of $q$ is equal to number of axes in the root system of corresponding Lie group. It is also valid for Steinberg groups. The exception is Suzuki and Ree groups $^2B_2, ^2G_2, ^2F_4$ where this number is $2,3,12$ which is half of number of axes in corresponding root system.
Is it possible to find representation of root system in the finite group in following way. Axis in root system is represented by element of order $q$. Number $q=p^l$ is power of certain prime $p$. Perpendicular axes correspond to commuting elements. Not perpendicular axes correspond to not commuting elements.
I admit that maybe elements of order $p$ can be used. I tested groups $L_3(2)$ and $U_4(2)$ and I found respectively $3$ and $6$ involutions having described property. Three involutions generated whole group $L_3(2)$. Six involutions generated subgroup of size $2^6$ in $U_4(2)$. Product of not commuting involutions have order 4.
Next question is what subgroup such "root system" of elements will generate. I guess it should be either Sylow subgroup of size $q^k$ or full group.
End of question
In Robert Wilson's book about finite simple groups I can find phrase "maximal torus" used when discussing subgroups of $F_4(q)$ and $E_6(q)$. This means that we can use terms from Lie groups in finite groups of Lie type. My next goal would be defining multiplication in the group by using the root system and number $q=p^l$.
Yet another goal is to see how we can understand sporadic group by looking at its p-Sylow group for certain prime p, whose power at least 3 divide order of the group.
End of motivation