Exercise 17, § 9

(a) Let $c$ denote a real solvable Lie algebra with a basis $(a, b, c)$ such that $[a, b] = c, [a, c] = -b, [b, c] = 0$. Let $d$ denote a real solvable Lie algebra with a basis $(a, b, c, d)$ such that $[a, b] = c, [a, c] = -b, [b, c] = d, [a, d] = [b, d] = [c, d] = 0$. Let $g$ be a real solvable Lie algebra. If $g$ contains non-zero elements $x, y, z$ such that $[x, y] = z$ and $[x, z] = -y$, then $g$ contains a subalgebra isomorphic to $c$ or $d$. (Write $a_1 = y, b_1 = z, c_1 = [y, z]$; define $a_i, b_i, c_i$ inductively by $a_i = [a_{i-1}, c_{i-1}], b_i = [b_{i-1}, c_{i-1}], c_i = [a_i, b_i]$. Consider the smallest integer $k$ such that $c_k = 0$.)

(b) Let $g$ and $g^*$ be real solvable Lie algebras and $\phi$ a homomorphism of $g$ onto $g^*$. If $g^*$ contains a subalgebra isomorphic to $c$ or $d$, $g$ has the same property.

(c) Let $h$ be a complex solvable Lie algebra. Consider a Jordan–Hölder series for the adjoint representation of $h$; the quotients of this series define 1-dimensional representations of $h$ and hence linear forms on $h$. These linear forms, which depend only on $h$, are called the roots of $h$. If $h'$ is a real solvable Lie algebra, the roots of $h'$ are the restrictions to $h'$ of the roots of $h' \otimes_{\mathbf{R}} \mathbf{C}$.

Then let $G$ be a finite-dimensional simply connected solvable real Lie group and $g$ its Lie algebra. Show that the following conditions are equivalent: ($\alpha$) $\exp_G$ is injective; ($\beta$) $\exp_G$ is surjective; ($\gamma$) $\exp_G$ is bijective; ($\delta$) $\exp_G$ is an isomorphism of the analytic manifold $L(G)$ onto the analytic manifold $G$; ($\varepsilon$) $L(G)$ contains no subalgebra isomorphic to $c$ or $d$; ($\zeta$) there exists no quotient algebra of $L(G)$ admitting a subalgebra isomorphic to $c$; ($\eta$) every root of $g$ is of the form $\phi + i\phi'$ where $\phi, \phi'$ are in $g^*$ and $\phi'$ is proportional to $\phi$; ($\theta$) for all $x \in g$, the only pure imaginary eigenvalue (in $\mathbf{C}$) of $\mathrm{ad}\, x$ is 0.\footnote{For more details, cf. M. Saito, Sur certains groupes de Lie résolubles, Sci. Papers of the College of General Education, Univ. of Tokyo, 7 (1957), pp. 1–11 and 157–168.}