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  "body": "\n## Summary\n\nA Lie subgroup of a [[Lie group|7JBRGCVJRB]] $G$ is a [[subgroup|0X1JXT47GA]] with its own Lie group structure.\nThe inclusion into $G$ must be a smooth [[immersion|6RW68JGBHC]].\nIts topology may be finer than the subspace topology, so a Lie subgroup need not be closed or embedded.\nThe connected Lie subgroups of $G$ correspond exactly to the Lie subalgebras of the Lie algebra of $G$.\n\n## Explanations\n\n### L1 human\n\nA Lie subgroup is a smaller Lie group sitting inside a bigger one.\nIt can be a nice closed piece, like the rotations about one axis inside all rotations of space.\nIt can also wind around in a complicated way.\nIn a torus, a line with irrational slope wraps around forever and comes close to every point, yet it is a Lie subgroup, a copy of $\\R$.\nIts own smooth structure is that of the line, not the one it would inherit as a subset of the torus.\n",
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