Existence of a regular semisimple element over $\mathbb{F}_{q}$

Question

This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help.

Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{F}_{q}$. Is there a clean reference in the literature to the fact that there must be at least one regular semisimple element in $G(K)$?

I know from the literature that regular semisimple elements are dense in $G$ and in every maximal torus $T$ (for example §2.3 and §2.5 in Humphreys's Conjugacy classes in semisimple algebraic groups), but this ensures only that there are many elements over the algebraic closure $\overline{K}$, not specifically over $K$. I know that this is true for $\mathrm{char}(K)=0$, using unirationality and the fact that $K$ is infinite, as in this question. I even know how to presumably shoot my question with a cannon in a few different ways, for instance:

1. Simply connected (almost) simple groups are in 1-1 correspondence with Dynkin diagrams (see for instance §32 in Humphreys's Linear algebraic groups), so I could manually write down one regular semisimple element over $\mathbb{F}_{q}$ for each such $G$. [EDIT: as pointed out in the comments below by LSpice, since the field in not algebraically closed there's no 1-1 correspondence as stated. One needs to consider automorphisms of the Dynkin diagrams as well.]
2. Fleischmann and Janiszczak (1993) have formulas for the number of regular semisimple elements over $\mathbb{F}_{q}$ at least for the classical $G$, i.e. of type $A,B,C,D$; they even announce on p. 484 that they obtained the number of semisimple conjugacy classes for $E$, so possibly they cover those cases as well. In any case, if there is a formula and its result is $>0$, then in particular there is one such element.
3. Lehrer (1992) showed among other things the "curious" result that, independently of characteristic, any semisimple simply connected $G$ has an odd number of regular semisimple conjugacy classes in $G(K)$ (see Corollary 3.5). Again, in particular if the number is odd then it is not $0$.

I would classify each of the routes above as "too strong for my question". I feel there must be some chapter-1-of-a-textbook kind of reference for "there exists a regular semisimple element over $\mathbb{F}_{q}$", but I could not find one.

Do you know where to find it? Also, bonus content: is the fact true and referenced if I relax the hypotheses (connected, or semisimple, or reductive...)?

Answer 1

See Proposition 7.1.4 in Dat, Orlik, and Rapoport, Period domains over finite and $p$-adic fields, Cambridge tracts in Mathematics, vol. 183. While I don't have this book in front of me, I'm pretty sure that the result asserts the existence of elliptic semisimple elements in all finite reductive groups. Such elements are necessarily regular. Most groups are handled via a uniform argument, but several are treated as special cases. Technically, the proof contains a gap, as the case of $G_2(\mathbb{F}_2)$ was inadvertently omitted. Nonetheless, the result remains true in this case, where one can find a regular element by looking at the Coxeter torus $T$ of $\operatorname{SL}_3$, which we can consider as a subgroup of $G_2$. Every nontrivial element of $T(\mathbb{F}_2)$ is regular in $G_2$.

Answer 2

(Edit: the previous version of this answer was not correct, but I leave this here as a remark)

Let $\sigma: G \rightarrow G$ be the Frobenius map corresponding to the field automorphism $x \mapsto x^q$.

To show that there exists a regular semisimple element in $G_{\sigma} = G(q)$, by the Lang-Steinberg theorem it would be enough to show that there is a $\sigma$-invariant class of regular semisimple elements. See 2.7(a) in "Conjugacy Classes" by Springer and Steinberg (in Lecture Notes in Math 131).