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In the ordered basis $(K_1,K_2,J)$, the three Adjoint representation matrices areTheir cross-product traces vanish, while their squared traces are $-2\lambda$, $-2\lambda$ and $-2{\lambda }^2$. ConsequentlyThis is the Killing form for cyclic three-generator brackets with constants $(1,\lambda ,\lambda )$.

For $\lambda >0$, the form is nondegenerate and negative-definite. The algebra is semisimple and compact, isomorphic to the SU(2) Lie algebra. For $\lambda <0$, it is nondegenerate with signature $(2,1)$, so the algebra is semisimple and noncompact, isomorphic to ${\mathfrak{sl}}_2(\mathbb{R})$.

For $\lambda =0$, the only nonzero bracket is $[K_1,K_2]=J$, with $J$ central. This is the Heisenberg Lie algebra, which is nilpotent and neither semisimple nor compact. Compactness is ruled out even though the Killing form is merely degenerate: $A_{K_1}$ is nonzero and nilpotent, whereas in a compact Lie algebra every adjoint map is skew-adjoint for a positive inner product. Skew-adjoint nilpotent operators vanish.