A new construction of $\tilde{D}_5$-singularities and generalization of Slodowy slices

Any simple elliptic singularity of type $\tilde{D}_5$ can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional "good slices" in the semi-simple Lie algebra ${\frak sl}(2, {\mathbb C}) \oplus {\frak sl}(2, {\mathbb C})$. We describe these new slices purely by the structure of the Lie algebra. We also construct the semi-universal deformation spaces of $\tilde{D}_5$-singularities by using the 4-dimensional "good slices".