Real form of a Lie group

Summary

A real form of a complex Lie group $G_\C$ is a closed real Lie subgroup $G$. Its Lie algebra must be a real form of the Lie algebra of $G_\C$. So $G$ has half the real dimension of $G_\C$, and complexifying its Lie algebra gives back that of $G_\C$. One complex group has several real forms, for example $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$ in $\mathrm{SL}_n(\C)$. The classification of real semisimple groups is largely the classification of real forms of complex semisimple groups.