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  "body": "\n## Statement\n\nLet $G$ be a connected [[semisimple Lie group|XJS07AJ50Y]] with finite center and Lie algebra $\\mathfrak{g}$.\nLet $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}$ be a Cartan decomposition, as in the [[Cartan decomposition of a Lie group|A5JB9JQKR0]].\nLet $K$ be the connected subgroup with Lie algebra $\\mathfrak{k}$.\nLet $\\mathfrak{a} \\subseteq \\mathfrak{p}$ be a maximal abelian subspace, and let $A = \\exp \\mathfrak{a}$.\nLet $N_K(\\mathfrak{a})$ and $Z_K(\\mathfrak{a})$ be the normalizer and the centralizer of $\\mathfrak{a}$ in $K$ under the adjoint action.\nLet $W = N_K(\\mathfrak{a}) / Z_K(\\mathfrak{a})$.\nThen $G = KAK$: each $g \\in G$ can be written $g = k_1 \\exp(H) k_2$ with $k_1, k_2 \\in K$ and $H \\in \\mathfrak{a}$.\nIn such a product, $H$ is unique up to the action of $W$ on $\\mathfrak{a}$.\n\n## Remarks\n\nThe idea of the proof: by the Cartan decomposition, $g = k \\exp X$ with $X \\in \\mathfrak{p}$.\nEach element of $\\mathfrak{p}$ is conjugate under $\\mathrm{Ad}(K)$ to an element of $\\mathfrak{a}$, so $X = \\mathrm{Ad}(k_2^{-1}) H$ and $\\exp X = k_2^{-1} \\exp(H) k_2$.\nUniqueness comes from the fact that two elements of $\\mathfrak{a}$ conjugate under $\\mathrm{Ad}(K)$ are conjugate under the Weyl group $W$.\n\nExample: $G = \\mathrm{SL}_n(\\R)$, $K = \\mathrm{SO}(n)$, $A$ the positive diagonal matrices of determinant $1$.\nThe theorem is the singular value decomposition $g = k_1 \\operatorname{diag}(\\sigma_1, \\dots, \\sigma_n) k_2$.\nHere $W$ is the symmetric group $S_n$, and the singular values are unique up to order.\n\nThe factors $k_1$ and $k_2$ are not unique: for $g = e$ one may take $k_1 = k$, $H = 0$, $k_2 = k^{-1}$ for any $k \\in K$.\n\n## Explanations\n\n### L1 human\n\nEvery element of a semisimple Lie group is a rotation, then a stretching along fixed axes, then another rotation.\nFor matrices this is the singular value decomposition.\nThe amounts of stretching are well defined up to the order of the axes.\n\n### L2 human\n\nThe decomposition is the group form of the fact that $K$ acts on the symmetric space $G/K$ and each orbit meets the flat $A \\cdot o$.\nIt gives polar coordinates on $G$.\nIn these coordinates the Haar measure has a density in $H$, a product of powers of $\\abs{\\sinh \\lambda(H)}$ over the positive restricted roots $\\lambda$.\nThis is the starting point of harmonic analysis on $G/K$ and of the study of spherical functions.\n",
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