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Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra. [ABSTRACT FROM AUTHOR]