The Euclidean algebra in rank 2 classical Lie algebras

We classify the embeddings of the Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(2)$\end{document}e(2) into the rank 2 semisimple, classical Lie algebras. The classifications are up to inner automorphism. We also examine the finite-dimensional, irreducible representations of the rank 2 semisimple, classical Lie algebras restricted to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(2)$\end{document}e(2) with respect to various embeddings. All Lie algebras and representations are over the complex numbers.

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