We explicitly determine all Rota-Baxter operators (of weight zero) on |$\mathrm{sl(2,\mathbb {C})}$| under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in |$\mathrm{sl(2,\mathbb {C})}$|, confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the six-dimensional Lie algebra |$\mathrm{sl(2,\mathbb {C})}\ltimes _{{\rm ad}^{\ast }} \mathrm{sl(2,\mathbb {C})}^{\ast }$|. They also give rise to three-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other six-dimensional Lie algebras.
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