I need a reference to a complete list of all
faithful real 4-dimensional irreducible representations
of real Lie algebras.


The list itself is not very hard to obtain.
Using the Levi decomposition,
it's possible to see that the Lie algebra $\mathfrak g
\subset\mathfrak{sl}(4, {\Bbb R})$ which does not
fix a proper subspace has to be reductive.
Then (up to a center acting by constants and/or 
by rotations with constant angle),
$\mathfrak g$ is a semisimple subalgebra
of $\mathfrak{sl}(4, {\Bbb R})=\mathfrak{so}(3,3)$.
The list of such subalgebras (I think) is
$\mathfrak{so}(1,2)=\mathfrak{sl}(2, {\Bbb R})$,
$\mathfrak{so}(3)$, $\mathfrak{so}(2,2)=
\mathfrak{so}(1,2)\times \mathfrak{so}(1,2)$,
$\mathfrak{so}(4)=\mathfrak{so}(3)\times \mathfrak{so}(3)$,
$\mathfrak{so}(1,3)=\mathfrak{sl}(2, {\Bbb C})$,
$\mathfrak{so}(2,3)=\mathfrak{sp}(4, {\Bbb R})$,
$\mathfrak{sl}(4, {\Bbb R})=\mathfrak{so}(3,3)$.
It can be (probably) obtained by removing some vertices of
the Dynkin diagram of $A\_3$ and using the arcane technique of coloring the remaining vertices to obtain the real forms. All in all, it's a pain.


I am sure there is a reference somewhere
to this list (also I might have missed some
algebras). I would be much grateful for all pointers.



A list of all irreducible 4-dimensional real representations
A complete list of semisimple subalgebras of $\mathfrak{sl}(4,\mathbb{R})$ seems to be available here: <http://downloads.hindawi.com/journals/jmath/2016/2570147.pdf>


