Subalgebras of Lie algebras. Example of 𝔰𝔩3(ℝ) revisited

Revisiting the results by Winternitz [Subalgebras of Lie algebras. Example of [Formula: see text], in Symmetry in Physics, CRM Proceedings and Lecture Notes, Vol. 34 (American Mathematical Society, Providence, RI, 2004), pp. 215–227], we thoroughly refine his classification of Lie subalgebras of the real order-three special linear Lie algebra and thus present the correct version of this classification for the first time. A similar classification over the complex numbers is also carried out. We follow the general approach by Patera, Winternitz and Zassenhaus but in addition enhance it and rigorously prove its theoretical basis for the required specific cases of classifying subalgebras of real or complex finite-dimensional Lie algebras. As a byproduct, we first construct complete lists of inequivalent subalgebras of the rank-two affine Lie algebra over both the real and complex fields.