This question is related to one asked earlier about inductive presentations of unipotent radicals in Kac-Moody groups.

Let $\Gamma$ be a *coxeter diagram* --- i.e. an unoriented graph with $r$ vertices whose edges are labeled by the integer weights $\big\{3,4,6 \big\}$. Let $\vec{\Gamma}$ be a directed version of $\Gamma$ where each edge with weight $4$ or $6$ is assigned an orientation. Let $W$ and $G$ respectively denote the coxeter group and the Kac-Moody group associated to $\Gamma$ and $\vec{\Gamma}$; in addition choose an opposing pair of Borel subgroups $B_{\pm}$ of $G$ and let $N_{\pm}$ denote the corresponding unipotent radicals.
Given $w \in W$ let $\dot{w}$ denote a lifting of $w$ to $G$ and
consider the subgroup $N(w) := N \cap \big( \dot{w} \big)^{-1} N_{-} \big( \dot{w} \big)$ of $N$; it is a finite dimensional nilpotent group which,
as a variety, is isomorphic to $\Bbb{C}^{l(w)}$ where $l(w)$ is the length
of $w$.

**Question** Let $c = \sigma_1 \dots \sigma_r$ be a choice of coxeter element of $W$ and consider $N\big(c^2 \big)$. Is there way to read off
a presentation of this group from $\Gamma$; in a manner analagous
to the Chevalley presentation of $N$ (but with added relations) ?

regards, A. Leverkühn

p.s. This question only becomes interesting in the case that $l \big(c^2 \big) = 2 \cdot l(c)$ otherwise $N \big( c^2 \big)$ will be abelian. For example type $\Gamma = A_r$ is **not** interesting
while type $\Gamma = D_r$ for $r \geq 4$ is. Is it possible that $N \big( c^2 \big) \cong \ N\big( c \big) \rtimes N \big( c \big)$ when this length condition is satisfied ?