A Lie algebra is called simple if it has no proper ideals and is not abelian. A Lie algebra is called semi-simple if it has no proper solvable ideals and is not abelian.

Let or . Examples of simple algebras are , the Lie algebra of the special linear group (traceless matrices), , the Lie algebra of the special orthogonal group (skew-symmetric matrices), and the Lie algebra of the symplectic group. Over , there are other simple Lie algebas, such as , the Lie algebra of the special unitary group (skew-Hermitian matrices). Any semi-simple Lie algebra is a direct product of simple Lie algebras.

Simple and semi-simple Lie algebras are one of the most widely studied classes of algebras for a number of reasons. First of all, many of the most interesting Lie groups have semi-simple Lie algebras. Secondly, their representation theory is very well understood. Finally, there is a beautiful classification of simple Lie algebras.

Over , there are 3 infinite series of simple Lie algebras: , and , and 5 exceptional simple Lie algebras , and . Over the picture is more complicated, as several different Lie algebras can have the same complexification (for example, and both have complexification ).