  Determining the structure of the Kauffman bracket skein module of all
$3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ is a
big open problem in skein theory. Very little is known about the skein module
of non-prime manifolds over this ring. In this paper, we compute the Kauffman
bracket skein module of the $3$-manifold $(S^1 \times S^2) \ \# \ (S^1 \times
S^2)$ over the ring $\mathbb Z[A^{\pm 1}]$. We do this by analysing the
submodule of handle sliding relations, for which we provide a suitable basis.
Along the way we also compute the Kauffman bracket skein module of $(S^1 \times
S^2) \ \# \ (S^1 \times D^2)$.
