On Nash resolution of (singular) Lie algebroids

Any Lie algebroid $A$ admits a Nash-type blow-up $\mathrm{N}\mathrm{a}\mathrm{s}\mathrm{h}(A)$ that sits in a nice short exact sequence of Lie algebroids $0\to K\to \mathrm{N}\mathrm{a}\mathrm{s}\mathrm{h}(A)\to \mathcal{D}\to 0$ with $K$ a Lie algebra bundle and $\mathcal{D}$ a Lie algebroid whose anchor map is injective on an open dense subset. The base variety is a blowup determined by the singular foliation of $A$. We provide concrete examples. Moreover, we extend the construction following Mohsen’s to singular subalgebroids in the sense of Androulidakis-Zambon.