= Killing form for cyclic three-generator brackets
{c}
{title2=$B=\operatorname{diag}(-2ac,-2ab,-2bc)$}
For $[e_1,e_2]=ae_3$, $[e_2,e_3]=be_1$, and $[e_3,e_1]=ce_2$, 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.
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