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

We show that there are four chiral $$ \mathcal{W} $$ W -algebra extensions of $$ \mathfrak{so}\left(2,3\right) $$ 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 $$ \mathfrak{sl}\left(2,\mathbb{R}\right) $$ sl 2 ℝ in the $$ \mathfrak{so}\left(2,3\right) $$ so 2 3 algebra and the maximal subalgebra $$ \mathfrak{h} $$ h that commutes with it. Then using the standard 2d chiral CFT techniques we find the corresponding $$ \mathcal{W} $$ W -algebra extensions. Two of the four resultant $$ \mathcal{W} $$ W -algebras are new, one of which may be thought of as the conformal $$ {\mathfrak{bms}}_3 $$ bms 3 algebra valid for finite values of its central charge.