Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{gl}_N(k)$ be (the $k$-points of) its Lie-algebra.

Is anybody aware of an accessible reference where I could find a classification of the orbits for the adjoint action of $G$ on $\mathfrak{g}$, including orbit sizes?

Other exceptional groups over $k$ are also of interest to me, so if such a reference for them also exists I'd be happy to hear about it.

Thank you!

Finite groups of Lie typealso has this -- see the tables in the back. (And don't confuse this book with his other book with titleSimple groups of Lie type.) $\endgroup$ – Nick Gill Jul 12 '18 at 17:31