  Any Lie algebroid $A$ admits a Nash-type blow-up $\mathrm{Nash}(A)$ that sits
in a nice short exact sequence of Lie algebroids $0\rightarrow K\rightarrow
\mathrm{Nash}(A)\rightarrow \mathcal{D}\rightarrow 0$ 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.
