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  "body": "\n## Example\n\nEvery compact connected [[simple Lie group|AM7YJP8AR6]] is isomorphic to a quotient $\\tilde{G}/C$.\nHere $\\tilde{G}$ is exactly one group of the following list, and $C$ is a subgroup of the centre of $\\tilde{G}$.\nEach $\\tilde{G}$ is compact, connected and simply connected, and its centre is finite.\n\n| Type | $\\tilde{G}$ | Dimension | Centre |\n|---|---|---|---|\n| $A_n$, $n \\ge 1$ | $SU(n+1)$ | $n(n+2)$ | $\\Z/(n+1)$ |\n| $B_n$, $n \\ge 2$ | $\\mathrm{Spin}(2n+1)$ | $n(2n+1)$ | $\\Z/2$ |\n| $C_n$, $n \\ge 3$ | $\\mathrm{Sp}(n)$ | $n(2n+1)$ | $\\Z/2$ |\n| $D_n$, $n \\ge 4$ | $\\mathrm{Spin}(2n)$ | $n(2n-1)$ | $\\Z/4$ for $n$ odd, $\\Z/2 \\times \\Z/2$ for $n$ even |\n| $G_2$ | $G_2$ | $14$ | trivial |\n| $F_4$ | $F_4$ | $52$ | trivial |\n| $E_6$ | $E_6$ | $78$ | $\\Z/3$ |\n| $E_7$ | $E_7$ | $133$ | $\\Z/2$ |\n| $E_8$ | $E_8$ | $248$ | trivial |\n\nHere $SU(n+1)$ is the [[special unitary group|CZCWFPNS4N]], $\\mathrm{Spin}(m)$ is the [[spin group|JJSXXJQGEH]], and $\\mathrm{Sp}(n)$ is the [[compact symplectic group|5NNPBFQA02]].\nThe groups [[G2|YX5FFSCR8P]] and [[E8|MKT1263M0A]] are the smallest and the largest exceptional groups.\nThe type is the type of the Dynkin diagram of the Lie algebra, and $n$ is the rank.\nCheck of one row: $SU(n+1)$ has Lie algebra of trace-zero skew-Hermitian $(n+1) \\times (n+1)$ matrices, of dimension $(n+1)^2 - 1 = n(n+2)$.\n\n## Remarks\n\nThe bounds on $n$ avoid repetitions and non-simple cases: $\\mathrm{Sp}(1) \\cong \\mathrm{Spin}(3) \\cong SU(2)$, $\\mathrm{Sp}(2) \\cong \\mathrm{Spin}(5)$, $\\mathrm{Spin}(6) \\cong SU(4)$, and $\\mathrm{Spin}(4) \\cong SU(2) \\times SU(2)$ is not simple.\nThe first four rows are the classical families, and the last five are the exceptional groups.\nFor example, $SO(3) = SU(2)/\\set{\\pm I}$ is the quotient for $A_1$ with $C$ the whole centre.\nThe same list of types classifies the complex simple Lie algebras.\nThe noncompact simple Lie groups are classified through the real forms of these complex Lie algebras, a classification due to Élie Cartan.\n\n## Explanations\n\n### L1 human\n\nThere are only a few kinds of compact simple Lie groups.\nFour infinite families come from matrices that preserve lengths over the real numbers, the complex numbers or the quaternions.\nBesides them there are exactly five exceptional groups, of dimensions $14$, $52$, $78$, $133$ and $248$.\nEvery compact simple Lie group is one of these, possibly divided by a finite group of central elements.\n",
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