  We give an explicit basis $\mathcal{B}$ of the quotient of the Kauffman
bracket skein algebra $\mathcal{S} (\Sigma)$ on a surface $\Sigma$ by the
square of an augmentation ideal. As an application, it induces two kinds of
finite type invariants of links in a handle body in the sense of Le. Moreover,
we construct an embedding of the mapping class group of a compact connected
surface of genus $0$ into the Kauffman bracket skein algebra on the surface
completed with respect to a filtration coming from the augmentation ideal.
