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!