A real Lie algebra is compact when it is the Lie algebra of some compact Lie group. Equivalently, it has a positive-definite invariant bilinear form on a Lie algebra. It is a direct sum of an abelian center and a compact semisimple Lie algebra. For its semisimple part, the compactness criterion from the Killing form is negative-definiteness of the Killing form. This definition concerns the algebra; a global group with that algebra can still have a noncompact central covering direction.
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