= Positive invariant metric on a Lie algebra
A positive invariant metric is a positive-definite real symmetric <invariant bilinear form on a Lie algebra>. Each adjoint map is skew-adjoint. Invariance gives $[\mathfrak g,\mathfrak g]^\perp=\mathfrak z(\mathfrak g)$, so the algebra splits orthogonally into its Abelian center and its derived algebra. The latter has zero center, and $\kappa(X,X)=\operatorname{tr}(\operatorname{ad}_X^2)=-\|\operatorname{ad}_X\|^2$ is strictly negative for nonzero $X$. Thus it is compact semisimple; the whole algebra is compact reductive. Conversely, a compact semisimple summand has metric $-\kappa$ and the center can carry any positive metric. The <Killing form> alone vanishes on the center.
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