The Killing form identifies roots with coroot directions

The Killing form of ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{C})$ is $B(X,Y)=2n\mathrm{tr}(XY)$, so for $H_{\alpha }=\frac{1}{2n}(E_{ii}-E_{jj})$ and a diagonal traceless $H$ with coordinates $x_k$ one gets $B(H_{\alpha },H)=\mathrm{tr}((E_{ii}-E_{jj})H)=x_i-x_j=\alpha (H)$. Hence $H_{\alpha }$ is exactly the Killing-dual vector of $\alpha$ from Killing-dual vector of a root, and the map $H\mapsto B(H,\cdot )$ is an isomorphism onto ${\mathfrak{h}}^{\ast }$ because $B{\mathrm{\mid }}_{\mathfrak{h}}$ is nondegenerate by Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra.

Since $E_{ii}-E_{jj}$ is traceless diagonal, $\alpha (E_{ii}-E_{jj})=2$, and therefore $\alpha (H_{\alpha })=\frac{1}{2n}\cdot 2=\frac{1}{n}$, which is nonzero as required by The Killing length of a root is nonzero and agrees with the direct computation $B(H_{\alpha },H_{\alpha })=2n\mathrm{tr}\left(\frac{(E_{ii}-E_{jj}{)}^2}{4n^2}\right)=\frac{1}{n}$.

Hence $h_{\alpha }=2H_{\alpha }\mathrm{/}\alpha (H_{\alpha })=2n{H}_{\alpha }=E_{ii}-E_{jj}$ by Coroot of a Lie-algebra root, and the coroot triple $e_{\alpha }=E_{ij}$, $f_{\alpha }=E_{ji}$, $h_{\alpha }=E_{ii}-E_{jj}$ is the one computed in The root sl_2 triple inside sl_n; in particular the coroot direction is the dual direction of the root under the Killing form, and the normalization is exactly $\alpha (h_{\alpha })=2$.