Solution
= Solution
In <Dynkin label> coordinates, subtracting $\alpha_i$ subtracts row $i$ of the stated Cartan matrix. Starting from $(1,0,0)$ gives the weight chain
$$
(1,0,0)
\xrightarrow{-\alpha_1}(-1,1,0)
\xrightarrow{-\alpha_2}(0,-1,1)
\xrightarrow{-\alpha_3}(0,0,-1).
$$
These are the four weights of the defining representation of $A_3\cong\mathfrak{sl}_4$.