the following product (in the universal enveloping algebra $\mathfrak U(\mathfrak u^-)$) $$f=f_{{\alpha}_i}\cdots f_{{\alpha}_1}\cdot f_{{\alpha}_{i+1}}\cdots f_{{\alpha}_{r-1}}\cdot f_{{\alpha}_r}\cdots f_{{\alpha}_2},$$ and consider all the factorizations $f = p\cdot q$ such that $p,q \in\mathfrak U(\mathfrak u^-)$. If ${\beta}_1,\ldots,{\beta}_j\in{\Delta}$, set $$\,^\mathrm r(f_{{\beta}_1}\cdots f_{{\beta}_j})=(-1)^j2^\delta f_{{\beta}_j}\cdots f_{{\beta}_1},$$ where $\delta$ equals 0 (resp. 1) if $\alpha_i$ occurs an even (resp. odd) number of times in $\{{\beta}_1,\ldots,{\beta}_j\}$. Then it is easy to check that the vector $$\sum_{p\cdot q=f} p.v_0 \otimes \,^\mathrm r\!q.w_0$$ is a $U$-invariant vector in ${V({\omega_2})}\otimes{V({\omega_i})}$ of $T$-weight $\omega_i$.

If the Dynkin diagram of $G$ is not simply laced we will need some further properties of tensor products.

If ${\Delta}$ is connected but not simply laced, we will denote by $\alpha_S$ the short simple root that is adjacent to a long simple root $\alpha_L$; moreover, we will denote the associated fundamental weights by $\omega_S$ and $\omega_L$. Finally, define $\zeta$ as the sum of all simple roots and notice that $\omega_S + \zeta$ is dominant.

\[lem:zeta\] Let ${\lambda}$ be a non-zero dominant weight.

1.  If $G$ is of type ${\mathsf F}_4$ or ${\mathsf C}_r$ ($r{\geqslant}3$) and if $\operatorname{Supp}({\lambda})$ contains a long root then $${V({{\lambda}+\omega_S})} \subset {V({\zeta + \omega_S})} \otimes {V({{\lambda}})}.$$

2.  If $G$ is of type ${\mathsf G}_2$ 