Real Lie subalgebras of o(3,1)

Question

I'm interested in the (nonabelian) real subalgebras of the Lorentz algebra, in particular those of dimension 4. I couldn't find a reference discussing this problem. Any guidance would be appreciated.

Answer 1

There's been a lot of work done classifying subalgebras of semisimple Lie algebras. The work is most often dealing with the complex case, but considerable work has been done in the real case. Originally, mostly semisimple subalgebras were considered, but more recently solvable and Levi decomposable subalgebras have been studied.

The subalgebras of $\mathfrak{so}(3,1)$ were examined in the article "Lie subalgebras of $\mathfrak{so}(3,1)$ up to conjugacy" by Ghanam, Thompson, and Bandara. The article specifically includes a section on the 4-dimensional subalgebras that you're interested in. The article identifies a Borel subalgebra (unique up to conjugacy) as the only 4-dimensional subalgebra. They also identify this Lie algebra with respect to Snobl and Winternitz's classification in their text "Classification and identification of Lie algebras."