The paper [*Invariants of real low dimension Lie algebras*][1] ([journal link][2]) 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.


  [1]: http://www.ams.org/mathscinet-getitem?mr=404362
  [2]: http://aip.scitation.org/doi/10.1063/1.522992