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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center, with Lie algebra $\\mathfrak{g}$ and Killing form $B$.\nLet $\\theta$ be a Cartan involution of $\\mathfrak{g}$: an automorphism with $\\theta^2 = \\id$ such that $(X, Y) \\mapsto -B(X, \\theta Y)$ is positive definite.\nLet $\\mathfrak{k}$ and $\\mathfrak{p}$ be the eigenspaces of $\\theta$ for the eigenvalues $1$ and $-1$, so that $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}$.\nLet $K$ be the connected Lie subgroup of $G$ with Lie algebra $\\mathfrak{k}$.\nThen:\n\n1. $K$ is a compact subgroup of $G$, and it is a [[maximal compact subgroup|T2EB8R9HEV]];\n2. the map $K \\times \\mathfrak{p} \\to G$, $(k, X) \\mapsto k \\exp X$, is a diffeomorphism, where $\\exp$ is the [[exponential map|H3QX1ZFC8K]].\n\nIn particular $G = K \\exp(\\mathfrak{p})$.\n\n## Remarks\n\nA Cartan involution exists and is unique up to inner automorphisms; this is the Cartan decomposition of the Lie algebra.\nThe proof of 2 lets $G$ act on the symmetric space $G/K$, which has nonpositive curvature.\nThen the map $X \\mapsto \\exp(X) K$ is a diffeomorphism from $\\mathfrak{p}$ onto $G/K$.\n\nExample: $G = \\mathrm{SL}_n(\\R)$ with $\\theta(X) = -X^{T}$.\nThen $\\mathfrak{k} = \\mathfrak{so}(n)$, $\\mathfrak{p}$ is the space of symmetric matrices of trace $0$, and $K = \\mathrm{SO}(n)$.\nThe theorem is the polar decomposition: each $g \\in \\mathrm{SL}_n(\\R)$ is uniquely $g = k e^{X}$ with $k$ orthogonal and $e^X$ symmetric positive definite.\n\nWithout a finite center, part 2 still holds with $K$ closed, but $K$ is no longer compact.\nFor the universal cover of $\\mathrm{SL}_2(\\R)$, the subgroup $K$ is isomorphic to $\\R$.\n\n## Explanations\n\n### L1 human\n\nEvery element of a semisimple Lie group splits as a rotation part times a stretching part.\nThe rotation part lies in a compact subgroup $K$.\nThe stretching part is the exponential of an element of a vector space $\\mathfrak{p}$.\nFor invertible matrices this is the familiar fact that a matrix is a rotation times a positive symmetric matrix.\n\n### L2 human\n\nThe theorem shows that $G$ is diffeomorphic to $K \\times \\R^{d}$ with $d = \\dim \\mathfrak{p}$.\nSo the topology of $G$, for example its fundamental group, is the topology of the compact group $K$.\nThe quotient $G/K$ is a Riemannian symmetric space of noncompact type, diffeomorphic to $\\mathfrak{p}$.\nThe Iwasawa decomposition $G = KAN$ and the decomposition $G = KAK$ refine this splitting.\n",
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