Killing form of the planar Euclidean motion Lie algebra ($[\kappa ]=\mathrm{diag}(-2,0,0)$)

In the ordered basis $(J,E_1,E_2)$ of the Euclidean motion Lie algebra in two dimensions, the Killing form has matrix $\mathrm{diag}(-2,0,0)$. Its radical is the translation ideal, and the algebra is solvable rather than semisimple.