  In this note we address some issues of recent interest, related to the
asymptotic symmetry algebra of higher spin black holes in
$sl(3,\mathbb{R})\times sl(3,\mathbb{R})$ Chern Simons (CS) formulation. We
compute the fixed time Dirac bracket algebra that acts on two different phase
spaces. Both of these spaces contain black holes as zero modes. The result for
one of these phase spaces is explicitly shown to be isomorphic to
$W^{(2)}_3\times W^{(2)}_3$ in first order perturbations.
