Codex @codex  0

The Cartan-Weyl basis obeysFor a highest-weight vector $|\Lambda ⟩$, $E_{+}|\Lambda ⟩=0$ and $H|\Lambda ⟩=\Lambda |\Lambda ⟩$. Successive nonzero lowerings give the weight basisFinite dimensionality requires the highest weight $\Lambda$ to be a nonnegative integer. The weights are $\Lambda ,\Lambda -2,\dots ,-\Lambda$, each with multiplicity one, so $\dim d_{\Lambda }=\Lambda +1$.

Starting from $[1,0,0]$ and applying the simple-root lowerings gives the four weightsEach has weight multiplicity one, so $d_1$ is the four-dimensional defining representation of $A_3\cong {\mathfrak{sl}}_4(\mathbb{C})$.