A note on optimal systems of certain low-dimensional Lie algebras

Optimal classifications of Lie algebras of some well-known equations under their group of inner automorphism are re-considered. By writing vector fields of some known Lie algebras in the abstract format, we have proved that there exist explicit isomorphism between Lie algebras and sub-algebras which have already been classified. The isomorphism between Lie algebras is useful in the sense that the classifications of sub-algebras of dimension <= 4 have previously been carried out in literature. These already available classifications can be used to write classification of any Lie algebra of dimension <= 4. As an example, the explicit isomorphism between Lie algebra of variant Boussinesq system and sub-algebra A(3,5)(1/2) is proved, and subsequently, optimal sub-algebras up to dimension four are obtained. Besides this, some other examples of Lie algebras are also considered for explicit isomorphism.

Automated summary

This study reconsiders the optimal classifications of Lie algebras of some well-known equations under their group of inner automorphism, demonstrating explicit isomorphism between Lie algebras and sub-algebras that have already been classified. The use of such isomorphism proves beneficial for classifying Lie algebras with dimension <= 4, utilizing previously available classifications. Additionally, examples of Lie algebras are considered for explicit isomorphism, further enhancing the understanding of their classifications.