The paper Invariants of real low dimension Lie algebras (journal link) lists all real Lie algebras of dimension $\leq 5$ and all nilpotent of dimension $\leq 6$ along with its invariants. It also contains references to the papers where the classifications are obtained: work of Mubarakzyanov and Morozov.

The paper Invariants of real low dimension Lie algebras (journal link) lists all real Lie algebras of dimension $\leq 5$ and all nilpotent of dimension $\leq 6$ along with its invariants. It also contains references to the papers where the classifications are obtained: work of Mubarakzyanov and Morozov.