As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book *DG, Lie groups and symmetric spaces* and Knapp's *Beyond an introduction*.
$\newcommand{\Real}{{\mathbb R}}\newcommand{\fa}{{\sf a}}\newcommand{\fg}{{\sf g}}\newcommand{\fk}{{\sf k}}\newcommand{\fn}{{\sf n}}$


Let $G$ be a connected, semisimple, non-compact Lie group (over the reals, **not necessarily a matrix group**, and **not necessarily with finite centre**), let $\fg=\fk\oplus \fa\oplus\fn$ be an Iwasawa decomposition of its Lie algebra, and let $G=KAN$ be the corresponding "global" Iwasawa decomposition. Since $G$ is non-compact, both $A$ and $N$ are non-trivial; moreover, both are in fact closed, simply connected subgroups of $G$.


It is observed in Helgason's book that if $\fg$ is the Lie algebra of $G$ and $\fn$ is the nilpotent subalgebra constructed in the Iwasawa decomposition at the Lie algebra level, then for each $x\in \fn$ we can find $h\in\fg$ such that $[h,x]=x$. (This is one of the steps in the proof of the Jacobson-Morozov theorem on existence of copies of ${\sf sl}(2,\Real)$ inside $\fg$.)



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> Q1. Can we always find $x\in\fn$, $x\neq 0$, and $h\in\fa$, such that $[h,x]=x$?
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I suspect that one could modify Jacobson's argument (*Proc. AMS* 1951) or the version Helgason gives, due to Kostant (*Amer. J. Math.* 1959, Section 3) to do this, but I have not succeeded in nailing down an argument.


If the answer to Q1 is positive, then I think we should be able to exponentiate $h$ and $x$ and obtain a **closed** and **simply connected** subgroup of $AN$, which would then be isomorphic to the so-called $ax+b$ group ${\Real}\rtimes {\Real}\_+^\*$. (Of course one can see this directly for $SL(2,\Real)$ and thence for $SL(n,\Real)$, which is the case that motivated the present question.)



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> Q2. Assuming it is true that $AN$ always contains a closed copy of the $ax+b$ group, is this standard knowledge for which there exists a reasonably precise citation?
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Copies of ax+b inside the AN part of an Iwasawa decomposition?
Here is an elementary argument addressing Q1. The adjoint action of the subalgebra $\mathfrak{a}$ on $\mathfrak{g}$ has the following two key properties:


1. It is diagonalizable (the operators $ad(H)$ for $H\in\mathfrak{a}$ are symmetric with respect to the Killing form);
2. It stabilizes $\mathfrak{n}$.


It follows that either $\mathfrak{a}$ centralizes $\mathfrak{n}$ or there exists an element $H\in\mathfrak{a}$ with a non-zero eigenvalue. In the latter case, rescaling, we get a pair $H\in\mathfrak{a}, X\in\mathfrak{n}$ such that $[H,X]=ad(H)X=X.$ As Yves has already remarked, the former case is never realized since the Lie algebra $\mathfrak{g}$ is semisimple. 


