The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: \Bbb{C} \longrightarrow \text{N}$ subject to the following defining relations, namely:
\begin{equation} \begin{array}{llc} x_1(a)\, x_2(b) \, x_1(c) &= \ x_2 \Bigg( \displaystyle {bc \over {a+c}} \Bigg) \, x_1 \Big(a + b\Big) \, x_2 \Bigg( {ab \over {a+c}}\Bigg) &\text{for type $A_2$} \\ \\ x_1(a) \, x_2(b) \, x_1(c) \, x_2(d) &= \ x_2 \Bigg(\displaystyle {bc^2d \over {\pi_2}} \Bigg) \, x_1 \Bigg( {\pi_2 \over {\pi_1}} \Bigg) \, x_2 \Bigg( {\pi_1^2 \over \pi_2} \Bigg) \, x_1 \Bigg( {abc \over \pi_1} \Bigg)&\text{for type $B_2$} \\ \\ &\text{where} \ \pi_1 = ab + \big(a +c \big)d & \\ &\text{and} \ \pi_2 = a^2b + \big(a +c \big)^2d & \end{array} \end{equation}
For $G_2$ a six term relation is valid which converts a factorisation $x_1(a)x_2(b)x_1(c)x_2(d)x_1(e)x_2(f)$ into a product of the form $x_2(A)x_1(B)x_2(C)x_1(D)x_2(E)x_1(F)$ where $A$, $B$, $C$, $D$, $E$, and $F$ are subtraction-free rational expressions in the complex parameters $a$, $b$, $c$, $d$, $e$, and $f$ --- see the paper "Total positivity in Schubert varieties" by Arkady Berenstein and Andrei Zelevinsky.
Using a truncated version of the $\text{SL}_2 \Big( \Bbb{C} \Big)$ loop-group, namely $\text{SL}_2 \Big( \mathcal{L_d} \Big)$ where $\mathcal{L_d}$ is the Auslander ring $\Bbb{C}\big[ t \big] \Big/ \Big( t^d = 0 \Big)$ one can check the existence of similar relations of length $2d$ for $d \geq 1$; the cases of $d=1$ and $d=2$ correspond to types $B_2$ and $G_2$ respectively. Hint: interpret the abstract one-parameter groups $x_1$ an $x_2$ as the $\mathcal{L}_d$-valued $2 \times 2$ matrices
\begin{equation} x_1(a) = \begin{pmatrix} 1 & a \\ 0 & 1\end{pmatrix} \qquad x_2(a) = \begin{pmatrix} 1 & 0 \\ at & 1 \end{pmatrix} \end{equation}
These relations correspond to coxeter relations of length $2d$ within the group of dihedral symmetries of a regular $2d$-gon.
Question: Is there a length five relation of this kind ? If so, is there a nice representation of the corresponding nilpotent group ?
best, A. Leverkühn