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        "quote": "A <u class=\"concept-term\" data-concept-identity=\"urn:uuid:f0104c54-899b-429c-808b-f923ff1a7f81\" data-concept-route=\"/concept-identities#concept-f0104c54-899b-429c-808b-f923ff1a7f81\" id=\"first-15e2e825d739e6fbadb8-en\">Lie subgroup</u> $H$ of a Lie group $G$ is a subgroup that is a Lie group, and such that the natural injection map $i : H \\to G$ defined by $i ( g ) = g$ makes it into an embedded submanifold of $G$. It is called a <u class=\"concept-term\" data-concept-identity=\"urn:uuid:03eb86c3-86f8-40cb-8e28-d7d053422767\" data-concept-route=\"/concept-identities#concept-03eb86c3-86f8-40cb-8e28-d7d053422767\" id=\"first-0be35a567ad49081ee79-en\">closed subgroup</u> if in addition $H$ is a closed subset of $G$. In this case the embedding is regular and its topology is that induced by the topology of $G$ (see <a href=\"../chapter-15/04-tangent-map-and-submanifolds#example-15-12\" class=\"cross-reference\" id=\"xref-03-lie-subgroups-1\" data-reference=\"example-15-12\">Example 15.12</a>). The injection $i$ induces a Lie algebra homomorphism $i _ { * } : { \\mathcal { H } } \\to ( G )$ , which is clearly an isomorphism of $\\mathcal{H}$ with a Lie subalgebra of $\\mathcal{G}$. We may therefore regard the Lie algebra of the Lie subgroup as being a Lie subalgebra of $\\mathcal{G}$.",
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