A list of all irreducible 4-dimensional real representations

Question

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.

Answer 1

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

Answer 2

Your list of semisimple subalgebras is not quite complete. You are missing (at least) $\mathfrak{sl}(3,\mathbb R)$ and $\mathfrak{sl}(2,\mathbb R)\times \mathfrak{sl}(2,\mathbb R)=\mathfrak{so}(2,2)=\mathfrak{so}(2,1)\times \mathfrak{so}(1,2)$. See Marcel Berger
Les espaces symétriques noncompacts, 1957, MR0104763.