To what extent, the following types of Lie algebras are classified :



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> Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact. 
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A question on Lie algebras
They are exactly the semi-simple Lie algebras which admit a positive definite invariant bilinear form (or equivalently, for which the Killing form is negative definite). An excellent reference for this is the first section of Bourbaki, *Lie groups and Lie algebras*, Chapter 9 (you need very little background to be able to read it).


