Editing Lie algebra (section)

== Real form and complexification ==
Given a [[complex Lie algebra]] <math>\mathfrak g</math>, a real Lie algebra <math>\mathfrak{g}_0</math> is said to be a ''[[real form]]'' of <math>\mathfrak g</math> if the complexification <math>\mathfrak{g}_0 \otimes_{\mathbb R} \mathbb{C}</math> is isomorphic to <math>\mathfrak{g}</math>. A real form need not be unique; for example, <math>\mathfrak{sl}(2,\mathbb{C})</math> has two real forms up to isomorphism, <math>\mathfrak{sl}(2,\mathbb{R})</math> and <math>\mathfrak{su}(2)</math>.<ref name="Fulton 26">{{harvnb|Fulton|Harris|1991|loc=§26.1.}}</ref>

Given a semisimple complex Lie algebra <math>\mathfrak g</math>, a ''[[split form]]'' of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphism). A ''[[compact form]]'' is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique up to isomorphism.<ref name="Fulton 26" />