  This paper introduces an algebra structure on the part of the skein module of
an arbitrary $3$-manifold $M$ spanned by links that represent $0$ in
$H_1(M;\mathbb{Z}_2)$ when the value of the parameter used in the Kauffman
bracket skein relation is equal to $\pm {\bf i}$. It is proved that if $M$ has
no $2$-torsion in $H_1(M;\mathbb{Z})$ then those algebras, $K_{\pm {\bf
i}}^0(M)$, are naturally isomorphic to the corresponding algebras when the
value of the parameter is $\pm 1$. This implies that the algebra $K_{\pm{\bf
i}}^0(M)$ is the unreduced coordinate ring of the variety of
$PSL_2(\mathbb{C})$-characters of $\pi_1(M)$ that lift to
$SL_2(\mathbb{C})$-representations.
