Real form

Summary

A real form of a complex Lie algebra $\mathfrak{g}$ is a real Lie subalgebra $\mathfrak{g}_0$ whose complexification is $\mathfrak{g}$, that is, $\mathfrak{g} = \mathfrak{g}_0 \oplus i\mathfrak{g}_0$. One complex Lie algebra can have several non-isomorphic real forms. A real simple Lie algebra is a real form of a complex simple Lie algebra or a complex simple Lie algebra seen as real. So the classification of real simple Lie algebras reduces to the classification of real forms.