  The symmetrized polydisc of dimension three is the set \[ \Gamma_3 =\{
(z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3
\} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which
$\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. We show that
every $\Gamma_3$-contraction admits a decomposition into a $\Gamma_3$-unitary
and a completely non-unitary $\Gamma_3$-contraction. This decomposition
parallels the canonical decomposition of a contraction into a unitary and a
completely non-unitary contraction. We also find new characterizations for the
set $\Gamma_3$ and $\Gamma_3$-contractions.
