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  "body": "\n## Summary\n\nA simple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a simple Lie algebra.\nSuch a Lie algebra is not abelian and has no ideals other than $0$ and itself.\nSimple Lie groups are the building blocks of the [[semisimple Lie groups|XJS07AJ50Y]], and through them of a large part of Lie theory.\nUp to coverings and central quotients, they are classified by the simple Lie algebras.\nThese form the classical families $A_n, B_n, C_n, D_n$ and five exceptional types.\n\n## Explanations\n\n### L1 human\n\nA simple Lie group is a continuous group of symmetries that cannot be broken into smaller continuous pieces.\nThink of the rotations of space: there is no smaller family of rotations, closed under conjugation and depending continuously on parameters, other than the trivial one.\nSimplicity is checked on the infinitesimal level, on the Lie algebra, so a simple Lie group may still have a finite centre.\nFor example, $SU(2)$ is a simple Lie group even though its centre $\\set{I, -I}$ is a normal subgroup.\nThe rotation group $SO(3)$, the groups $SL(n, \\R)$ for $n \\ge 2$ and the exceptional group $E_8$ are other examples.\n",
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