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>



Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below
Consider the nilpotent group $N={\mathbb R} \rtimes {\mathbb R}^2$ (the group of $3\times 3$ upper triangular unipotent matrices with real coefficients. If $v,w \in {\mathbb R}^2$, then their commutator $[v,w]$ in $N$ is simply the wedge $v\wedge w \in \wedge ^2 {\mathbb R}^2\simeq {\mathbb R}$. 


The group $H=SL(2,{\mathbb R})$ operates on $N$ since it preserves the symplectic form $v\wedge w$ on ${\mathbb R}^2$. 


