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  "body": "\n## Statement\n\nLet $G$ be a connected [[compact Lie group|9TFC6ADC1C]] and let $T$ be a [[maximal torus|3JVGZVVTB0]] of $G$.\n\n1. For every maximal torus $T'$ of $G$ there is $g \\in G$ with $T' = gTg^{-1}$.\n2. Every element of $G$ lies in some maximal torus; equivalently, $G = \\bigcup_{g \\in G} gTg^{-1}$.\n\n## Remarks\n\nOne proof uses the Lefschetz fixed point theorem on the manifold $G/T$.\nFor $x \\in G$, the map $gT \\mapsto xgT$ of $G/T$ is homotopic to the identity, since $G$ is connected.\nSo its Lefschetz number is the Euler characteristic of $G/T$, which equals the order of the Weyl group and is nonzero.\nA fixed point $gT$ means $xgT = gT$, that is, $g^{-1}xg \\in T$; this gives part 2, and part 1 follows by taking for $x$ a generator of a dense cyclic subgroup of $T'$.\nExample: for $G = U(n)$ and $T$ the diagonal unitary matrices, part 2 is the spectral theorem: every unitary matrix is $g D g^{-1}$ with $g$ unitary and $D$ diagonal.\nConnectedness is needed for part 2: in $O(2)$, every torus lies in the identity component $SO(2)$, so a reflection lies in no maximal torus.\n\n## Explanations\n\n### L1 human\n\nIn a compact connected Lie group there is essentially only one maximal torus: all of them are rotated copies of each other.\nMoreover these copies cover the whole group.\nFor unitary matrices this is the familiar fact that every unitary matrix becomes diagonal in a suitable orthonormal basis.\n\n### L2 human\n\nPart 2 implies that a continuous class function on $G$ is determined by its restriction to $T$, and that restriction is invariant under the [[Weyl group|EZGTEK3HQN]].\nThis is why characters of representations are studied on $T$, and why the [[Weyl integration formula|QMH530N27M]] reduces integrals of class functions to integrals over $T$.\nIt also implies that the exponential map of a compact connected Lie group is surjective, since it is surjective on each torus.\n",
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