 Solution

If the three simple roots are distinct and every pair of root spaces had nonzero Lie bracket, part (iv) would put all three connections in the Dynkin diagram. Their underlying Coxeter graph would contain a triangle, contradicting part (ii). Hence at least one pair satisfies $\boxed{[{\mathfrak{g}}_{{\alpha }_i},{\mathfrak{g}}_{{\alpha }_j}]=0}$. If two of the listed roots coincide, their common root space is one-dimensional, so its bracket with itself is already zero. Thus the conclusion also covers that case.