The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal

We give an explicit basis ℬ of the quotient of the Kauffman bracket skein algebra 𝒮(Σ) on a surface Σ by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus 0 into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.