A proper 1-cocycle on a Podleś sphere

We define $1$-cocycles on coideal $\ast$-subalgebras of CQG Hopf $\ast$-algebras and consider the condition for $1$-cocycles to extend to $1$-cocycles on Drinfeld double coideals. We construct a $1$-cocycle on a Podleś sphere, which extends to a $1$-cocycle on $S{L}_q(2,\mathbb{R})$ and prove that it is proper.