I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ without non-trivial fixed points. Please enlighten me.
P.S. I added the ergodic theory tag because I believe such groups are seen there.
This is also posted in stackexchange. https://math.stackexchange.com/questions/3037426/examples-of-non-abelian-simply-connected-nilpotent-lie-groups