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In this note we show by implementing the associativity conditions of OPEs interms of modes that there exist exactly two chiral algebras consisting of oneVirasoro operator $T(z)$, one $\mathfrak{sl}(2, {\mathbb R})$ current $J_a(z)$(for $a =0, \pm 1$), and a doublet $G_s(z)$ (for $s = \pm1/2$) of currentalgebra primaries of dimension 3/2. The simplest of these is the recentlyuncovered chiral extension of the Poincare algebra $\mathfrak{iso}(1,3)$. Thesecond is a chiral extension of the $\mathfrak{so}(2,3)$. The latter is a novel${\cal W}$-algebra that is analogous to the famous Bershadsky-Polyakov algebra${\cal W}_3^{(2)}$, and is expected to play a role in the AdS/CFTcorrespondence.

A Chiral ${\cal W}$-Algebra Extension of $\mathfrak{so}(2,3)$