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  "body": "\n## Definition\n\nLet $G$ be a [[compact Lie group|9TFC6ADC1C]].\nA *torus* in $G$ is a compact connected abelian Lie subgroup of $G$.\nEach such subgroup is isomorphic to the [[torus group|XNAWD33HN1]] $\\mathbb{T}^k$ for some $k \\ge 0$.\nA *maximal torus* of $G$ is a torus $T \\subseteq G$ such that every torus $T'$ with $T \\subseteq T' \\subseteq G$ satisfies $T' = T$.\n\n## Remarks\n\nExample: in $G = U(n)$, the subgroup $T$ of diagonal matrices $\\operatorname{diag}(z_1, \\dots, z_n)$ with $\\abs{z_j} = 1$ is a maximal torus.\nIt is a torus, isomorphic to $\\mathbb{T}^n$.\nLet $T' \\supseteq T$ be a torus.\nEach element of $T'$ commutes with a diagonal matrix with distinct entries, so it is diagonal; thus $T' = T$.\nNon-example: in $U(2)$, the subgroup $\\set{\\operatorname{diag}(z, 1) : \\abs{z} = 1}$ is a torus, but it is not maximal, since it lies in the diagonal torus $\\mathbb{T}^2$.\nThe dimension of a maximal torus is the rank of $G$; for $U(n)$ it is $n$, and for $SU(n)$ it is $n - 1$.\nEvery torus of $G$ lies in some maximal torus, by a dimension argument.\n",
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