When looking into sizes of finite simple group of "Lie type", I observed that power of the $q$ 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.

Question

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 $q$. Perpendicular axes correspond to commuting elements. Not perpendicular axes correspond to not commuting elements.

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

There are also alternating and sporadic groups which does not have any $q$ number assigned. When we have proper definition of "root system" in finite group then we can see if it can be extended somehow to sporadic or alternating groups.