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{0}^{\ \scriptscriptstyle T} & v \\ \hline \Bbb{O} & \vec{0} \\ \end{array} \right) \end{equation} for any vector $\vec{u}$ in $\Bbb{C}^{n-1}$ and any $v \in \Bbb{C}$. Again $\acute{H}$ is a normal and 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 $N_{B_{n-1}} \, \acute{H} = N_{B_n}$. Let $\acute{\mathcal{H}}$ denote the $\log$ of $\acute{H}$ so to speak --- the vector space $\Bbb{C}^{n-1} \oplus \Bbb{C}$ consisting of all tuples $\big(\vec{u}, v \big)$. The conjugation action of $N_{B_{n-1}}$ 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. yours as always, Ines