Consider triples $(\lambda,\mu,\nu)$ of dominant weights of $G$ such that the irrep $V_\nu$ occurs in $V_\lambda \otimes V_\mu$. Then this space of triples is closed under addition.

Proof. An intertwiner can be identified with a $G$-invariant section of the $(\lambda^*,\mu,\nu)$ equivariant line bundle over $(G/B)^3$. Tensoring two such sections together, we get a third, which is again nonzero because $(G/B)^3$ is reduced and irreducible.

(Moreover, this monoid is finitely generated, also not hard to prove with this approach.)