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  "body": "\n## Statement\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\\mathfrak{g}$ and [[exponential map|H3QX1ZFC8K]] $\\exp$.\nLet $H$ be a [[subgroup|0X1JXT47GA]] of $G$ that is a [[closed subset|0ZCK8DQYN1]] of $G$.\nThen $H$ is an embedded submanifold of $G$, and with this smooth structure $H$ is an embedded [[Lie subgroup|NM8XCPS1Q3]] of $G$.\nIts Lie algebra is\n$$\\mathfrak{h} = \\set{X \\in \\mathfrak{g} : \\exp(tX) \\in H \\text{ for all } t \\in \\R}.$$\n\n## Remarks\n\nThe proof first shows that $\\mathfrak{h}$ is a linear subspace, using the limits $\\exp(X + Y) = \\lim_n (\\exp(X/n)\\exp(Y/n))^n$ and closedness of $H$.\nThen one chooses a complement $\\mathfrak{m}$ of $\\mathfrak{h}$ in $\\mathfrak{g}$ and shows that $(Y, X) \\mapsto \\exp(Y)\\exp(X)$ on $\\mathfrak{m} \\times \\mathfrak{h}$ is a chart near $e$ in which $H$ is the slice $Y = 0$.\nThe key point is that $H$ has no elements of the form $\\exp(Y)$ with small nonzero $Y \\in \\mathfrak{m}$; otherwise a limit argument gives a nonzero element of $\\mathfrak{h} \\cap \\mathfrak{m}$.\nExample: $SO(n) = \\set{A \\in GL_n(\\R) : A^T A = I, \\det A = 1}$ is closed, so it is a Lie group; its Lie algebra is the space of skew-symmetric matrices.\nThe closedness hypothesis is necessary: the image of $t \\mapsto (e^{it}, e^{i\\sqrt2 t})$ in the torus $S^1 \\times S^1$ is a subgroup that is dense and not closed, and it is not an embedded submanifold.\n\n## Explanations\n\n### L1 human\n\nTo check that a group of matrices is a Lie group, one would normally need to build charts and check smoothness.\nThis theorem removes the work: any subgroup of a Lie group that is closed as a set is automatically a smooth submanifold and a Lie group.\nSo all the usual groups cut out by equations, such as orthogonal, unitary and symplectic groups, are Lie groups at once.\n\n### L2 human\n\nThe theorem is often stated for matrix groups and attributed to von Neumann in that case and to Élie Cartan in general.\nA typical use: the stabiliser $H$ of a point under a smooth action is closed, hence a Lie subgroup.\nThen the quotient $G / H$ is a manifold, a homogeneous space.\nThe formula for $\\mathfrak{h}$ is the practical way to compute Lie algebras: for $SL_n(\\R)$ it gives the matrices of trace $0$, because $\\det e^{tX} = e^{t \\tr X}$.\nIt also gives Cartan's theorem that a continuous homomorphism of Lie groups is smooth, by applying it to the graph.\n",
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