$\mathfrak{sl}(2,\mathbb{C})$ is indeed equal to $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R})$. The famous representation is given by $L_{-1}$, $L_0$, $L_1$ of the [Virasoro algebra](https://en.wikipedia.org/wiki/Virasoro_algebra) for the first copy of $\mathfrak{sl}(2, \mathbb{R})$, and by their conjugations $\overline{L}_{-1}$, $\overline{L}_0$, $\overline{L}_1$ for the second one.