  Let $G$ be a connected and simply connected complex semisimple Lie group. For
a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas
${\mathcal{A}}_{\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we
prove that all of its coordinate functions are positive with respect to the
Lusztig positive structure on $G/Q$. We also show that the standard Poisson
structure $\pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts
of ${\mathcal{A}}_{\rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a
certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q,
\pi_{G/Q}, {\mathcal{A}}_{\rm BS}(G/Q))$ into a Poisson-Ore variety. Examples
of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T \subset G$ is a
maximal torus, $B \subset G$ a Borel subgroup, and $N$ the uniradical of $B$.
