Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the *standard* Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie type over a field of $q$ elements and I note that all simple groups of Lie type can be obtained this way. 


(I am using `Frobenius map' in the sense of Carter as I want to include Ree and Suzuki groups.)


I believe the following to be true and would like help with a reference:


*If $q$ is large enough, then every torus in $G^F$ contains a regular semisimple element.* 


I would, moreover, like a precise definition of `large enough.' The statement and the definition of large enough can be derived from the following statement (although I'm less confident that this one is correct!):


*If a torus $T$ in $G^F$ is non-degenerate, then $T$ contains a regular semisimple element.*


(The statement now follows because if $q-1$ is greater than the height of the maximum root of $G$, then every torus of $G^F$ is non-degenerate. This result can be derived from the proof of Prop. 3.6.6 of Carter's "Finite Groups of Lie type")


If anyone can help with a proof/reference for either of the italicised statements I would be very grateful!



Regular elements in the torus of a group of Lie type
This question is interesting as it forces one to look closely at rational points of $F$-stable tori in $G$.


If $T$ is such torus then $F$ acts the lattice of cocharacters $X\_\*(T)$ thus inducing an automorphism of the set of coroots. Let $\sigma$ be this automorphism and suppose it has order $m$ (this is all we need to know about it). Let $\epsilon$ be a generator of the multiplicative group of the field in $q^m$ elements. Then for every coroot $\gamma^\vee$ we have that $\gamma^\vee(\epsilon)^F=(\sigma\gamma^\vee)(\epsilon^q)$. Now set 
$H\_\gamma:=\prod\_{i=0}^{m-1}(\sigma^i\gamma^\vee)(\epsilon^{q^i})$. Then $H\_\gamma^F=H\_\gamma$ by construction. Let $\alpha\_1,\ldots, \alpha\_l$ be a basis of simple roots in $X(T)$ and $a\_1,\ldots, a\_l\in\mathbb{Z}$. Define $H(a\_1,\ldots, a\_l):=H\_{\alpha\_1}^{a\_1}\cdots H\_{\alpha\_l}^{a\_l}$, an $F$-stable element of $T$. 


We want fo find $a\_i$ such that $H(a\_1,\ldots, a\_l)$ is regular (subject to some condition on $q$ to be determined). Let $a=\sum\_{i-1}^la\_i\alpha\_i^\vee$, an element of $X\_\*(T)$, and let $e\_\gamma\in g\_\gamma$, a root vector in $g=Lie(G)$. Set $\tau=\sigma^{-1}$. Then $(Ad\ H(a\_1,\ldots, a\_l))\cdot e\_\gamma=
\epsilon^{M(a)}e\_\gamma$ where $M(a)=\langle\gamma,\sum\_{i=0}^{m-1}q^i\sigma^i(a)\rangle=
\sum\_{i=0}^{m-1}q^i\langle\tau^i\gamma,a\rangle$.
Now $\epsilon$ is a primitive root of $1$ of order $q^m-1$. Suppose we found $a$ such that
 $\langle\beta,a\rangle\ne 0$ and
$-(q-1)< \langle\beta,a\rangle< q-1$ for all roots $\beta$. Then it follows from the uniqueness of $q$-adic expansions that $-(q^m-1)< M(a)< q^m-1$ and $M(a)\ne 0$, so that $H(a\_1,\ldots, a\_l)$ is regular as wanted.
The condition on $a$ will depend on $q$, of course, but NOT on the choice of $T$, and it is quite easy to make it explicit. 


Perhaps I should also mention that in the notation of Prop. 3.6.6 in Carter's book the condition $q\ge|\alpha|+1$ for all
$\alpha$ is not always tha same as $q-1>ht(\alpha)$ for all positive roots $\alpha$ (especially when there are roots of different lengths).


