Exercise 5, § 3

Equip $\mathbf{R}^n$ with the quadratic form $x_1^2 + x_2^2 + \cdots + x_n^2$. Let $G_n$ be the group of displacements of $\mathbf{R}^n$ (Alg., Ch. IX, §6, No. 6, Def. 3) of determinant 1. a) Show that $G_n$ is unimodular. b) Every element of $G_2$ may be uniquely written in the form

where $u, v$ are in $\mathbf{R}$, and $\omega$ is an element of the group $\Theta$ of angles of half-lines, isomorphic to $\mathbf{U}$. Show that $du \otimes dv \otimes d\omega$ (where $d\omega$ is a Haar measure on $\Theta$) is a Haar measure on $G_2$.