What you have asked for is indeed a fact in characteristic zero. See Corollary 8.5 of Borel-Tits (Publ. IHES at Numdam, unrestricted access).
However, it is false in positive characteristic. See also section 4 of the same article (where other fields are considered). Indeed, the group $\mathrm{PGL}_1(D)$ over a field $k$ of positive characteristic $p$, $D$ a central division algebra over $k$ of degree $p$ , can have "bad" unipotent elements coming from purely inseparable extensions of $k$ lying in $D$. What is true is that $G$ is anisotropic iff it does not have a proper parabolic subgroup defined over $k$.
In any case, this article by Borel and Tits is the standard reference for reductive groups over arbitrary fields.
