A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties
can be proved equivalent for a finite-dimensional algebra 
over a field of characteristic 0:
1. 
is semisimple.
2. 
has no nonzero Abelian ideal.
3. 
has zero ideal radical (the radical is the biggest
solvable ideal).
4. Every representation of 
is fully reducible, i.e., is a sum of irreducible representations.
5. 
is a (finite) direct product of simple Lie algebras
(a Lie algebra is called simple if it is not Abelian
and has no nonzero ideal 
).
