Examples of non-abelian simply connected nilpotent Lie groups.

Question

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.

Answer 1

The (real or complex) upper triangular group in size $\ge 3$ is a trivial example (with $H$ the diagonal group).