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