 Killing form for cyclic three-generator brackets

For $[e_1,e_2]=a{e}_3$, $[e_2,e_3]=b{e}_1$, and $[e_3,e_1]=c{e}_2$, the Adjoint representation matrices give the displayed Killing form. Cross traces vanish. Nonzero $a,b,c$ make it nondegenerate and hence the algebra semisimple. If the three constants have the same sign it is negative-definite; mixed signs give signature $(2,1)$. In particular $(a,b,c)=(2,2,2)$ is the compact $\mathfrak{su}(2)$ case, whereas $(2,2,-2)$ is the split real traceless two-by-two algebra. This gives a direct compactness criterion from the Killing form in a small explicit example.