Exercise 21, § 9

(a) Let G be a simply connected solvable real Lie group with the following properties: ($\alpha$) $L(G)$, which is n-dimensional, has a commutative ideal of dimension $n - 1$, corresponding to a subgroup A of G; ($\beta$) there exists an element $\sigma$ of the centre of G which does not belong to A. Show that there exists an element $x$ of $L(G)$ such that $\exp x = \sigma$. Show that $L(G)$ is the product of a commutative ideal and an ideal with a basis

such that $a_1, b_1, \ldots, a_k, b_k$ belong to $L(A)$, $[x, a_i] = 2\pi n_i b_i,\ [x, b_i] = -2\pi n_i a_i$ ($n_i \in \mathbf{Z} - \{0\}$) for all $i$. Generalize the results of Exercise 13 (d) to G.

(b) Let G be a finite-dimensional simply connected solvable real Lie group. Let D be a discrete subgroup of the centre of G. There exist a basis $(x_1, x_2, \ldots, x_n)$ of $L(G)$ and an integer $r \leq n$ with the following properties: (a) every element of G can be written uniquely in the form

where $t_1, \ldots, t_n$ are in $\mathbf{R}$; (β) $x_1, \ldots, x_r$ are pairwise permutable and is a basis of the commutative group D. (Argue by induction on the dimension of G. Let a be maximal commutative ideal of $L(G)$. Let A be the corresponding integral subgroup. Then A is closed in G and DA/A is a discrete subgroup of the centre of $G/A$, to which the induction hypothesis can be applied, whence there are elements $x_1^*, \ldots, x_m^*$ of $L(G/A)$ and an integer s. For $1 \leq i \leq s$, let $\sigma_i$ be an element of D whose class modulo A is $\exp x_1^*$. Construct one by one, using (a), representatives $x_1, \ldots, x_m$ of $x_1^*, \ldots, x_m^*$ such that $\exp x_i = \sigma_i$ and $[x_i, x_j] = 0$ for $1 \leq i, j \leq s.$\footnote{For more details, cf. C. Chevalley, Topological structure of solvable groups, Ann. of Math., 42 (1941), pp. 668–675.}

(c) Deduce from (b) that every finite-dimensional connected solvable real Lie group is homomorphic to a space $\mathbf{R}^n \times \mathbf{T}^m$ ($m, n$ integers $\geq 0$).