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 $\kappa (X,X)=\mathrm{tr}({\mathrm{ad}}_X^2)=-\Vert {\mathrm{ad}}_X{\Vert }^2$ 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.