 Solution

Use the A5 root system with simple roots ${\alpha }_i=e_i-e_{i+1}$ for $1\le i\le 5$, in the hyperplane ${\sum }_{i=1}^6{x}_i=0$. The Dynkin diagram is the five-node chainIts Cartan matrix isDeleting the final node leaves the A4 root system, with simple roots ${\alpha }_1,\dots ,{\alpha }_4$. Concretely, the Lie algebra homomorphism ${\mathfrak{sl}}_5(\mathbb{C})\hookrightarrow {\mathfrak{sl}}_6(\mathbb{C})$ sends $X$ to $\mathrm{diag}(X,0)$ and preserves the Lie bracket. This realizes the required Lie subalgebra and fixes the embedding used below.