On Nash resolution of (singular) Lie algebroids

Any Lie algebroid A admits a Nash-type blow-up Nash(A) that sits in a nice short exact sequence of Lie algebroids 0→ K→ Nash(A)→ D→ 0 with K a Lie algebra bundle and 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.