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

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.