We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

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Consider a simple complex Lie group G acting diagonally on a triple flag variety G/P 1 × G/P 2 × G/P 3 , where P i is parabolic subgroup of G. We provide an algorithm for systematically checking when this action has finitely many orbits. We then use this method to give a complete classification for when G is of type F 4 . The E 6 , E 7 , and E 8 cases will be treated in a subsequent paper.