  We use instanton gauge theory to prove that if $Y$ is a closed, orientable
$3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion
or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$
nor $\pm L(3,1)$, then there is an irreducible representation $\pi_1(Y) \to
\mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket
skein module of a non-prime 3-manifold has nontrivial torsion whenever two of
the prime summands are different from $\mathbb{RP}^3$, answering a conjecture
of Przytycki (Kirby problem 1.92(F)) unless every summand but one is
$\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show
that if $M$ is a compact, orientable $3$-manifold with torus boundary whose
rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto
the twisted $I$-bundle over the Klein bottle.
