Corollary 8.5 of Borel -Tits proves this in characteristic zero: 

 http://www.numdam.org/item?id=PMIHES_1965__27__55_0 

See also section 4 of the same article (where other fields are considered, but it is not said in terms of unipotent elements. Indeed, the group $PGL_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements.

In any case, this is the standard reference for reductive groups over arbitrary fields