= Compactness criterion from the Killing form
A finite-dimensional real <Lie algebra> has negative-definite <Killing form> exactly when it is compact semisimple. In the compact direction, average an <inner product> over a compact group; all adjoint maps are skew, so $\kappa(X,X)=-\|\operatorname{ad}_X\|^2$, and semisimplicity removes the center. Conversely, negative-definiteness gives the positive invariant metric $-\kappa$, embeds the faithful adjoint algebra in an orthogonal algebra, and nondegeneracy gives semisimplicity. This is an algebraic compactness criterion, not a claim that every global group with an Abelian compact-type algebra is compact.
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