 Solution

Put $H_{12}=E_{11}-E_{22}=[E_{12},E_{21}]$. The six roots of ${\mathfrak{sl}}_3$ take values $2,-2,1,-1,-1,1$ on $H_{12}$, so the root-space decomposition givesUsing the preceding invariant-form identity with $\alpha =L_1-L_2$,HenceThis agrees with the Killing form of the special linear Lie algebra, $B(A,C)=2n\mathrm{tr}(AC)$, at $n=3$.