All chiral ${\cal W}$-algebra extensions of $\mathfrak{so}(2,3)

We show that there are four chiral ${\cal W}$-algebra extensions of $\mathfrak{so}(2,3)$ algebra and construct them explicitly. We do this by a simple identification of each of the inequivalent embeddings of a copy of $\mathfrak{sl}(2,{\mathbb R})$ in the $\mathfrak{so}(2,3)$ algebra and the maximal subalgebra $\mathfrak{h}$ that commutes with it. Then using the standard 2d chiral CFT techniques we find the corresponding ${\cal W}$-algebra extensions. Two of the four resultant ${\cal W}$-algebras are new, one of which may be thought of as the conformal $\mathfrak{bms}_3$ algebra valid for finite values of its central charge.}