highest root if $\Phi$ is simply laced or the highest short root if $\Phi$ is not simply laced. For the convenience of the reader we list the highest short root of every irreducible root system in Table \[tab:hsr\].

   type of $\Phi$                                                highest short root
  ----------------- ------------------------------------------------------------------------------------------------------------
   ${\mathsf A}_r$                                ${\alpha}_1+\cdots+{\alpha}_r=\omega_1+\omega_r$
   ${\mathsf B}_r$                                    ${\alpha}_1+\cdots+{\alpha}_r=\omega_1$
   ${\mathsf C}_r$                      ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-1})+{\alpha}_r=\omega_2$
   ${\mathsf D}_r$              ${\alpha}_1+2({\alpha}_2+\cdots+{\alpha}_{r-2})+{\alpha}_{r-1}+{\alpha}_r=\omega_2$
   ${\mathsf E}_6$                ${\alpha}_1+2{\alpha}_2+2{\alpha}_3+3{\alpha}_4+2{\alpha}_5+{\alpha}_6=\omega_2$
   ${\mathsf E}_7$         $2{\alpha}_1+2{\alpha}_2+3{\alpha}_3+4{\alpha}_4+3{\alpha}_5+2{\alpha}_6+{\alpha}_7=\omega_1$
   ${\mathsf E}_8$   $2{\alpha}_1+3{\alpha}_2+4{\alpha}_3+6{\alpha}_4+5{\alpha}_5+4{\alpha}_6+3{\alpha}_7+2{\alpha}_8=\omega_8$
   ${\mathsf F}_4$                           ${\alpha}_1+2{\alpha}_2+3{\alpha}_3+2{\alpha}_4=\omega_4$
   ${\mathsf G}_2$                                       $2{\alpha}_1+{\alpha}_2=\omega_1$

  : []{data-label="tab:hsr"}

Recall the condition $(\star)$ defined in the introduction: a dominant weight ${\lambda}$ satisfies $(\star)$ if, for every non-simply laced connected component ${\Delta}'\subset {\Delta}$, if $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root then it contains also the short root which is adjacent to a long simple root.

\[twin\] If ${\Delta}' \subset {\Delta}$ is a non-simply laced connected component, order the simple roots in ${\Delta}'= \{ {\alpha}_1, \ldots,
{\alpha}_r\}$ starting from the extreme of the Dynkin diagram of ${\Delta}'$ which contains a long root and denote ${\alpha}_q$ the first short root in ${\Delta}'$. If ${\lambda}$ is a dominant weight such that ${\alpha}_q\not \in \operatorname{Supp}({\lambda})$ and such that $\operatorname{Supp}({\lambda})\cap {\Delta}'$ contains a long root, denote ${\alpha}_p$ the last long root which occurs in $\operatorname{Supp}({\lambda})\cap {\Delta}'$; for instance, if ${\Delta}'$ 