  A triple of commuting operators for which the closed tetrablock
$\overline{\mathbb E}$ is a spectral set is called a tetrablock contraction or
an $\mathbb E$-contraction. The set $\mathbb E$ is defined as \[ \mathbb E = \{
(x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever }
|z|\leq 1, |w|\leq 1 \}. \] We show that every $\mathbb E$-contraction can be
uniquely written as a direct sum of an $\mathbb E$-unitary and a completely
non-unitary $\mathbb E$-contraction. It is analogous to the canonical
decomposition of a contraction operator into a unitary and a completely
non-unitary contraction. We produce a concrete operator model for such a triple
satisfying some conditions.
