On the Steinberg property of the Contou-Carrere symbol

The aim of this work is to show that the Contou-Carr`{e}re symbol satisfies the property of Steinberg symbols: $<f,1-f{>}_{A((t){)}^{\times }}=1$ for all elements $f,1-f\in A((t){)}^{\times }$. Moreover, we offer a cohomological characterization of the Contou-Carr`{e}re symbol from the commutator of a central extension of groups.