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Oct 13, 2021 at 17:05 comment added მამუკა ჯიბლაძე Many thanks, great! In the meanwhile I also found examples with parabolic subgroups in $\operatorname{SL}_7$, $\operatorname{Sp}_8$ and $\operatorname{SO}_9$, but they are not nearly as clear to me as yours.
Oct 13, 2021 at 13:28 history edited YCor CC BY-SA 4.0
added example with nilpotent radical
Oct 13, 2021 at 9:50 comment added მამუკა ჯიბლაძე Nevertheless, it is THE answer :)
Oct 13, 2021 at 9:49 vote accept მამუკა ჯიბლაძე
Oct 13, 2021 at 8:57 comment added YCor (This answers your question. However this is not the nilpotent radical in the semidirect product.)
Oct 13, 2021 at 8:49 history answered YCor CC BY-SA 4.0