inductive construction of unipotent radicals Consider a directed coxeter diagram $\vec{\Gamma}$, i.e. a finite graph where each edge is decorated with one of the integer weights $\big\{3,4,6\big\}$ and those edges with weights $4$ or $6$ are assigned orientations.
Let $p$ be a vertex of $\vec{\Gamma}$ attached to a leaf $q$ and let
$m= m_{p,q}$ be the weight of the corresponding edge. Let $\mathring{\vec{\Gamma}}$ be the induced coxeter diagram obtained by removing the leaf $q$.
Let $G$ and $\mathring{G}$ be the Kac-Moody groups associated to 
$\vec{\Gamma}$ and $\mathring{\vec{\Gamma}}$ respectively. Choose Borel 
subgroups $B$ and $\mathring{B}$ of $G$ and $\mathring{G}$ and let
$N$ and $\mathring{N}$ denote their associated unipotent radicals.
Question: Is there a representation $\mathcal{H}_{p,m}$ of $\mathring{N}$ such that
$N \cong \mathcal{H}_{p,m} \rtimes \mathring{N}$ ?
 A: To A. Leverkühn,
This is not a general answer to your question but rather a nice confirmation in the case of $\vec{\Gamma} = B_n $. The dynkin diagram $B_n$ has two leaves
--- one joined by a type $A_2$-bond (weight $3$ by your convention) and another joined by a type $B_2$-bond (weight $4$ by your convention). So there
ought to be two distinct semi-direct product factorisations of the 
unipotent radical $N_{B_n}$ according to your conjecture. 
As I noted in your other posting the Kac-Moody group $G_{B_n}$ in this case is the symplectic group $\text{Sp}_{2n} \big( \Bbb{C} \big)$ whose unipotent radical $N_{B_n}$ can be identified as the group of all $2n \times 2n$ invertible complex matrices having the following block-decomposition
\begin{equation}
\left(
\begin{array}{c|c}
A & A \dot{w} S  \\ 
\hline  
\Bbb{O} &  \dot{w}A^{-T} \dot{w} \\
\end{array}
\right)
\end{equation}
where $A$ is any $n \times n$ unipotent matrix (i.e. upper-triangular with $1$'s on the diagonal), $S$ is any $n \times n$ symmetric matrix, and
$\dot{w}$ is the $n \times n$ permutation matrix of $w \in S_n$ 
defined by $w(i) = n+1 -i$ for $1 \leq i \leq n$. 
Consider first the case of the leaf attached by the $B_2$-bond. Let $H$
denote the abelian (indeed additive) subgroup of $N_{B_n}$ consisting of matrices of the form
\begin{equation}
 \left(
\begin{array}{c|c}
\Bbb{I} & \dot{w} S  \\ 
\hline  
\Bbb{O} &  \Bbb{I} \\
\end{array}
\right)
\end{equation}
It's easy to see that $H$ is normal in $N_{B_n}$; furthermore it is abelian and isomorphic to the vector space $\mathcal{H}$ consisting of all symmetric
complex matrices. The quotient $N_{B_n} \big/ H$ is isomorphic to the group
$N_{A_{n-1}}$ of all $n \times n$ unipotent complex matrices which acts
linearly on $\mathcal{H}$ by 
\begin{equation} A \cdot S \ = \ \dot{w} \, A \,\dot{w} \, S \, \dot{w} \, A^{T} \, \dot{w}
\end{equation}
and consequently we have the semi-direct product factorisation 
$N_{B_n} \cong \mathcal{H} \rtimes N_{A_{n-1}}$ expressed as
\begin{equation} \big(S_1,A_1\big) \circ \big(S_2,A_2 \big) \ = \ \Big(S_2 + A_2^{-1} \cdot S_1 \, , \,  A_1A_2 \Big) \end{equation}
Consider now the leaf with the type $A_2$-bond. In view of your conjecture we should expect a semi-direct factorisation involving $N_{B_{n-1}}$ instead of $N_{A_{n-1}}$. To effect this decomposition let's embed $\text{Sp}_{2n-2}\big( \Bbb{C} \big)$
into the central $(2n-2) \times (2n-2)$ block of $\text{Sp}_{2n} \big( \Bbb{C} \big)$, namely
\begin{equation}
\left(
\begin{array}{c|c|c}
1 &  &   \\ 
\hline 
 & \text{Sp}_{2n-2}\big( \Bbb{C} \big) &  \\
\hline 
 &  & 1
\end{array}
\right)
\end{equation}
Let $\acute{H}$ be the subgroup of $N_{B_n}$ consisting of matrices
having the block-form
\begin{equation}
\left(
\begin{array}{c|c}
U & V  \\ 
\hline  
\Bbb{O} & \dot{w}U^{-T}\dot{w} \\
\end{array}
\right)
\end{equation}
where 
\begin{equation}
U \ = \ \left( 
\begin{array}{c|l}
1 & \vec{u}^{ \ \scriptscriptstyle T}  \\ 
\hline  
\vec{0} & \Bbb{I} \\
\end{array}
\right) \quad \text{and} \quad 
V \ = \ \left(
\begin{array}{l|c} \vec{v}^{\ \scriptscriptstyle T} & v  \\ 
\hline  
\Bbb{O} & \vec{v} \\
\end{array}
\right)
\end{equation}
for any pair of vectors $\vec{u}$ and $\vec{v}$ in $\Bbb{C}^{n-1}$ and any
$v \in \Bbb{C}$. Again $\acute{H}$ is a normal (although non-abelian), it trivially intersects $N_{B_{n-1}}$ with respect to the embedding of $\text{Sp}_{2n-2}\big( \Bbb{C} \big)$ in $\text{Sp}_{2n}\big( \Bbb{C} \big)$, and $ N_{B_n} = \acute{H} \, N_{B_{n-1}}$. Let $\acute{\mathcal{H}}$ denote the $\log$ of $\acute{H}$ so to speak --- the vector space $\Bbb{C}^{n-1} \oplus \Bbb{C}^{n-1} \oplus \Bbb{C}$ consisting of all tuples $\big(\vec{u}, \vec{v}, v  \big)$. The conjugation action of $N_{B_{n-1}}$ on $\acute{H}$ induces a linear
action of $N_{B_{n-1}}$ on $\acute{\mathcal{H}}$ and so we get a second
semi-direct product factorisation $N_{B_n} \cong \acute{\mathcal{H}} \rtimes 
N_{B_{n-1}}$ as anticipated with the caveat however that although $\acute{\mathcal{H}}$ is a representation of $N_{B_{n-1}}$ its group structure
does not coincide with its additive structure as a vector space.
yours as always, Ines
