Some Leibniz bimodules of mathfrak{sl}₂

We study complex finite-dimensional Leibniz algebra bimodule over $\mathfrak{sl}_2$ that as a Lie algebra module is split into a direct sum of two simple $\mathfrak{sl}_2$-modules. We prove that in this case there are only two non-split Leibniz $\mathfrak{sl}_2$-bimodules and we describe the actions.