Gauge formulation of the spinning black hole in (2+1)-dimensional anti-de Sitter space

We compute the group element of SO(2,2) associated with the spinning black hole found by Ba\~nados, Teitelboim, and Zanelli in (2+1)-dimensional anti-de Sitter space-time. We show that their metric is built with SO(2,2) gauge invariant quantities and satisfies Einstein's equations with a negative cosmological constant. Moreover, although the metric is singular on the horizons, the group element is continuous and possesses a kink there. We also consider the extremal configurations and the Poincar\'e limit as special cases.