They are the compact Lie algebras. Note that are two different definitions in the literature. One is that a compact Lie algebra is the Lie algebra of a compact Lie group. This includes tori, and the Killing form then is negative semidefinite. The Lie algebra is not necessariliy semisimple, but reductive. The other definition is, that a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori.
Dietrich Burde
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