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What you have asked for is indeed a fact in characteristic zero. See Corollary 8.5 of Borel  -Tits:

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

However, it is false in positive characteristic. See also section 4 of the same article (where other fields are considered). 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 article by Borel and Tits is the standard reference for reductive groups over arbitrary fields

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.

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

What you have asked for is indeed a fact in characteristic zero. See Corollary 8.5 of Borel -Tits:

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

However, it is false in positive characteristic. See also section 4 of the same article (where other fields are considered). 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 article by Borel and Tits is the standard reference for reductive groups over arbitrary fields

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

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