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        "quote": "In addition, recall that this representation has a fourth isomorphic form: the representation of $\\mathrm{SO}(3)$ on column vectors. This is also a real representation, but it can be directly [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/complexification\" data-concept-identity=\"urn:uuid:5de2ca26-d059-42fc-9ea3-2beff8196917\" data-concept-route=\"/concept-identities#concept-5de2ca26-d059-42fc-9ea3-2beff8196917\" id=\"first-7d47e5ffbab5d81fba8b-en\">complexified</u>](../../../../../concepts/linear-algebra/complexification). Since $\\mathfrak{so}(3)$ and $\\mathfrak{su}(2)$ are isomorphic Lie algebras, their complexifications, $\\mathfrak{so}(3)_{\\mathbb{C}}$ and $\\mathfrak{sl}(2,\\mathbb{C})$, are also isomorphic.",
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