Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / d / i / Solution

In Dynkin label coordinates, subtracting ${\alpha }_i$ subtracts row $i$ of the stated Cartan matrix. Starting from $(1,0,0)$ gives the weight chainThese are the four weights of the defining representation of $A_3\cong {\mathfrak{sl}}_4$.