Any Lie algebroid A admits a Nash-type blow-up \({\textrm{Nash}}(A)\) that sits in a nice short exact sequence of Lie algebroids \(0\rightarrow K\rightarrow {\textrm{Nash}}(A)\rightarrow {\mathcal {D}}\rightarrow 0\) 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 blow-up determined by the singular foliation of A considered recently in Mohsen’s (Blow-up groupoid of singular foliations, 2021). We provide concrete examples. Moreover, we extend the construction to singular subalgebroids in the sense of Androulidakis–Zambon (Zambon in Annales de l’Institut Fourier 72(6):2109–2190, 2022).
