 Solution

For the usual positive roots, the chamber condition is that the coefficients of $L_1,L_2,L_3$ be weakly decreasing, since their consecutive differences are the simple-coroot pairings. Adding the same constant to all three coefficients changes no weight, because $L_1+L_2+L_3=0$.

The given coefficients are $(-2,3,0)$. Let $s_1$ exchange $L_1,L_2$, and $s_2$ exchange $L_2,L_3$. With the rightmost reflection acting first,ThusIts coefficients $(3,0,-2)$ are strictly decreasing, so it lies even in the open Weyl chamber; its Dynkin labels are $(3,2)$.