All chiral $\mathcal{W}$-algebra extensions of $\mathfrak{s}\mathfrak{o}(2,3)$

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