For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$,

we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,j}$ are the elements of the root subgroup $U_{\alpha_{i,j}}$)

I'm looking for sources that lists similar commutator relation in the diffrent root systems, and also the permutation of the longest Weyl element on the roots.

Specifically, I'm trying to find relations that enables me to move down the root element by hight using the highest root element and the longest Weyl element.

For example in $A_n$, $(\omega_0 x_{i-1,n}(t)\omega_0^{-1},x_{1,n}(u))=(x_{i,1}(t),x_{1,n}(u))=x_{i,n}(tu)$ ($\omega_0$ is the longest Weyl element and $x_{1,n}$ the highest root element).

page 3which I'm trying to expand. [1]: arxiv.org/abs/1511.07798 $\endgroup$ – Ami Aug 14 at 14:21