Exercise 10, § 9

Let G be the Lie subgroup of $\mathbf{G}\mathbf{L}(3,\mathbf{R})$ consisting of the matrices 1 $ab$

0 1 0 $,a, b\in \mathbf{R}$. Show that, for the identity representation of G on

0 0 1

$\mathbf{R}^3$, the number of orbit types is infinite.