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.