Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 1 / 4 / e

For a root $\alpha$, choose normalized root vectors $e_{\alpha }\in {\mathfrak{g}}_{\alpha }$, $f_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ and the coroot element $h_{\alpha }\in \mathfrak{h}$ withThe requested sl2 subalgebra associated with a root isThese bracket relations identify its basis with the standard sl2 triple. The notation here denotes this subalgebra, not the root reflection $s_{\alpha }$ used for the Weyl group.

The weight diagram of the Adjoint representation of ${\mathfrak{sl}}_3$ has its six roots $L_i-L_j$, $i\ne j$, each of multiplicity one, and zero of multiplicity two, from the Cartan subalgebra.

Restrict to the root subalgebra generated by $X=E_{12}$, $Y=E_{21}$ and $H=H_{12}$. The explicit decomposition into irreducible modules isThe first summand has $H$ weights $2,0,-2$. The second has weights $1,-1$, since $[Y,E_{13}]=E_{23}$. The third likewise has weights $1,-1$, since $[Y,E_{32}]=-E_{31}$. The last commutes with all three generators and is trivial. These subspaces are stable under the sl2 triple, are independent and have dimensions totaling eight, so they exhaust the Adjoint representation.

The Killing form value is the trace of $\mathrm{ad}{E}_{12}\mathrm{ad}{E}_{21}$. By the formula just proved, the four summands contribute $4,1,1,0$. Therefore