In many places one can find an information that real Lie algebras are classified up to dimension 5. The **only** reference containing a classification I was able to find is [Invariants of real low dimensional Lie algebras](http://aip.scitation.org/doi/10.1063/1.522992). The list presented there contains indecomposable Lie algebras of dimension up to five, yet every for and five dimensional algebra presented there is solvable. Is it true that there are no indecomposable, unsolvable algebras in that dimensions or the article **very implicitly** deals in fact only with solvable algebras.


An answer or a solid reference for a classification will be very appreciated.



Are there indecomposable unsolvable four and five dimensional Lie algebras?
Every finite dimensional real Lie algebra has the form $\mathfrak g=\mathfrak l\ltimes\mathfrak r$ where $\mathfrak l$ is semisimple and $\mathfrak r$ is solvable (Levi decomposition). The smallest semisimple Lie algebras are $\mathfrak{su}(1,1)=\mathfrak{sl}(2,\mathbb R)$ and $\mathfrak{su}(2)$, both of dimension $3$. The next larger ones have dimension $6$. So if $\mathfrak g$ is non-solvable of dimension $\le5$ then $\mathfrak l$ is one of the two $3$-dimensional ones. The indecomposability implies that $\mathfrak l$ acts non-trivially on $\mathfrak r$. From this, one gets easily that $\mathfrak{su}(1,1)$, $\mathfrak{su}(2)$, and $\mathfrak{sl}(2,\mathbb R)\ltimes\mathbb R^2$ are all the non-solvable indecomposable Lie algebras of dimension $\le5$. The last algebra is item $A\_{5,40}$ of Table II of loc. cit. which is incorrectly labeled as solvable.


