On triangles with a minuscule side

Abstract

Let $W\subset O(V)$ be the Weyl group of a root system $R\subset V$. If $a+b+c=0=a'+b'+c'$ with $a$, $b$ and $c$ respectively conjugated to $a'$, $b'$ and $c'$ in V , then $(a,b,c)$ is conjugated to $(a',b',c')$ in $V^3$ when each projection of $a$ to an irreducible component of $V$ is co-linear to a minuscule coweight.