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  "body": "\n## Definition\n\nLet $G_\\C$ be a complex [[Lie group|7JBRGCVJRB]] with complex Lie algebra $\\mathfrak{g}_\\C$.\nA *real form* of $G_\\C$ is a closed real [[Lie subgroup|NM8XCPS1Q3]] $G$ of $G_\\C$ whose Lie algebra $\\mathfrak{g} \\subseteq \\mathfrak{g}_\\C$ is a real form of $\\mathfrak{g}_\\C$.\nThat is, $\\mathfrak{g}_\\C = \\mathfrak{g} \\oplus i \\mathfrak{g}$ as real vector spaces.\n\n## Remarks\n\nExample: $G = \\mathrm{SU}(n)$ in $G_\\C = \\mathrm{SL}_n(\\C)$.\nHere $\\mathfrak{g} = \\mathfrak{su}(n)$ is the space of skew-Hermitian matrices of trace $0$, and $i \\mathfrak{g}$ is the space of Hermitian matrices of trace $0$.\nEach traceless complex matrix $Z$ is uniquely $\\frac{1}{2}(Z - Z^*) + \\frac{1}{2}(Z + Z^*)$, a skew-Hermitian plus a Hermitian traceless matrix, so $\\mathfrak{sl}_n(\\C) = \\mathfrak{g} \\oplus i\\mathfrak{g}$.\n\nNon-example: $G = \\mathrm{SO}(n)$ in $\\mathrm{SL}_n(\\C)$ for $n \\ge 2$.\nIts Lie algebra $\\mathfrak{so}(n)$ of real antisymmetric matrices satisfies $\\mathfrak{so}(n) \\cap i\\,\\mathfrak{so}(n) = 0$, but $\\mathfrak{so}(n) \\oplus i\\,\\mathfrak{so}(n)$ is the space of complex antisymmetric matrices, which misses the diagonal matrices of trace $0$.\n\nOften a real form is given as the fixed point group of an antiholomorphic involution $\\sigma$ of $G_\\C$, or as an open subgroup of it.\nFor $\\sigma(g) = \\bar g$ on $\\mathrm{SL}_n(\\C)$, the fixed points form $\\mathrm{SL}_n(\\R)$; for $\\sigma(g) = (g^*)^{-1}$, they form $\\mathrm{SU}(n)$.\nA real form that is compact is a compact real form; for a connected complex semisimple group it exists and is unique up to conjugation.\n",
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