Let $k$ be a finite field and $G$ a connected, reductive linear algebraic group defined over $k$. It is well-known that the union of the maximal tori of $G$ is dense in $G$ (more generally, if $G$ is not reductive, the union of its Cartan subgroups is dense).


It is also true, that $G$ is generated by the maximal tori defined over $k$ (this is proven in a paper by Borel and Springer from 1968). 


However, is the union of the maximal tori which are defined over $k$ still dense in $G$? 



Is the union of Cartan subgroups over $k$ dense?
The set of maximal tori defined over $k$ is finite: if $T$ is one of them, they correspond to the $k$-rational points of the $k$-variety $G/\text{(normalizer of }T)$. So their union is not dense, unless $G$ is a torus.


