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