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 $k=\R$ or $\C$ Examples of simple algebras are $\fr{sl}_nk$ the Lie algebra of the special linear group (traceless matrices), $\fr{so}_nk$ the Lie algebra of the special orthogonal group (skew-symmetric matrices), and $\fr{sp}_{2n} k$ the Lie algebra of the symplectic group. Over $\R$ there are other simple Lie algebas, such as $\fr{su}_n$ 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 $\C$ there are 3 infinite series of simple Lie algebras: $\fr{sl}_n$ $\fr {so}_n$ and $\fr{sp}_{2n}$ and 5 exceptional simple Lie algebras $\fr g_2,\fr f_4,\fr e_6,\fr e_7$ and $\fr e_8$ Over $\R$ the picture is more complicated, as several different Lie algebras can have the same complexification (for example, $\fr{su}_n$ and $\fr{sl}_n\R$ both have complexification $\fr{sl}_n\C$ .