Editing Lie algebra (section)

=== Classification ===
The [[Levi decomposition]] asserts that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its solvable radical and a semisimple Lie algebra.<ref>{{harvnb|Jacobson|1979|loc=Ch. III, § 9.}}</ref> Moreover, a semisimple Lie algebra in characteristic zero is a product of simple Lie algebras, as mentioned above. This focuses attention on the problem of classifying the simple Lie algebras.

The simple Lie algebras of finite dimension over an [[algebraically closed field]] ''F'' of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, using [[root system]]s. Namely, every simple Lie algebra is of type A<sub>''n''</sub>,  B<sub>''n''</sub>, C<sub>''n''</sub>,  D<sub>''n''</sub>, E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub>, F<sub>4</sub>, or G<sub>2</sub>.<ref>{{harvnb|Jacobson|1979|loc=section IV.6.}}</ref> Here the simple Lie algebra of type A<sub>''n''</sub> is <math>\mathfrak{sl}(n+1,F)</math>, B<sub>''n''</sub> is <math>\mathfrak{so}(2n+1,F)</math>, C<sub>''n''</sub> is <math>\mathfrak{sp}(2n,F)</math>, and D<sub>''n''</sub> is <math>\mathfrak{so}(2n,F)</math>. The other five are known as the [[exceptional Lie algebra]]s.

The classification of finite-dimensional simple Lie algebras over <math>\mathbb{R}</math> is more complicated, but it was also solved by Cartan (see [[simple Lie group]] for an equivalent classification). One can analyze a Lie algebra <math>\mathfrak{g}</math> over <math>\mathbb{R}</math> by considering its complexification <math>\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}</math>.

In the years leading up to 2004, the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic <math>p>3</math> were classified by [[Richard Earl Block]], Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See [[restricted Lie algebra#Classification of simple Lie algebras]].) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.