Reference Request: Anisotropic Algebraic Groups Have No Unipotent Elements

I have found the following fact stated in a number of places:

If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $mathrmHom_k(G, mathrmG_m)$ is trivial.

For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.

1 Answer
1

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, because that is false). 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 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 is the standard reference for reductive groups over arbitrary fields

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