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SOS-skeins correspond exactly to the Steiner quadruple systems [8,12]. Let \( P_1 \) be a finite simple SQS-skein of cardinality \( n > 4 \). In this article, we will present a construction for a non-simple subdirectly irreducible (monolithic) SOS-skein \( P = 2 \otimes_\alpha P_n \) of cardinality \( 2n \) in which each proper homomorphic image is Boolean for all \( n \equiv 2 \) or \( 4 \pmod{6} \). We can then show that if \( P_1 \) has a simple derived sloop, then the constructed SOS-skein \( 2 \otimes_\alpha P_1 \) contains a derived sloop which is subdirectly irreducible and has the same property as the SOS-skein \( 2 \otimes_\alpha P_1 \) that each of its proper homomorphic images is Boolean. Similar to the theory of Steiner loops and Steiner quasigroups [14], the author [1] has proven that the variety \( V(P_1) \) generated by a finite simple cubic SQS-skein \( P_1 \) covers the smallest non-trivial subvariety (the class of all Boolean SQS-skeins). Finally, we show that the variety \( V(2 \otimes_\alpha P_1) \) generated by the constructed SQS-skein \( 2 \otimes_\alpha P_1 \) covers the variety \( V(P_1) \) for each finite simple cubic SOS-skein \( P_1 \).