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  "body": "\n## Summary\n\nA real form of a complex Lie group $G_\\C$ is a closed real Lie subgroup $G$.\nIts Lie algebra must be a real form of the Lie algebra of $G_\\C$.\nSo $G$ has half the real dimension of $G_\\C$, and complexifying its Lie algebra gives back that of $G_\\C$.\nOne complex group has several real forms, for example $\\mathrm{SL}_n(\\R)$ and $\\mathrm{SU}(n)$ in $\\mathrm{SL}_n(\\C)$.\nThe classification of real semisimple groups is largely the classification of real forms of complex semisimple groups.\n\n## Explanations\n\n### L1 human\n\nA complex Lie group can be cut down to a real group in several ways.\nInside the complex $n \\times n$ matrices of determinant $1$, one can keep the matrices with real entries.\nOne can also keep the unitary ones, which preserve lengths in $\\C^n$.\nBoth choices have half the dimension, and both \"complexify\" back to the whole group.\nSuch real subgroups are the real forms.\n",
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