Exercise 4, § 3

Let $n$ be a natural number. Let $B_0$ be the subgroup of $\mathbf{S}\mathbf{L}(n,\mathbf{R})$ formed by the upper triangular matrices whose diagonal coefficients are all strictly positive.

(a) Exercise prove that the mapping from SO($n,\mathbf{R}$)$\times B_0$ into $\mathbf{S}\mathbf{L}(n,\mathbf{R})$ given by $(g, u)\mapsto g\cdot u$ is a homeomorphism (cf. INT, VII, p. 91, prop. 7).

(b) Exercise prove that the inclusion of SO($n,\mathbf{R}$) in $\mathbf{S}\mathbf{L}(n,\mathbf{R})$ induces an isomorphism between Poincaré groups.