Just a remark about the term compact Lie algebra. There are two different notions. One is that it is the Lie algebra of a compact Lie group. This includes tori, and the Killing form is negative semidefinite. Then 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.

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.