  A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed
tetrablock $\overline{\mathbb E}$ is a spectral set, is called a
\textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction},
where \[ \mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0
\quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3,
\] is a polynomially convex domain which is naturally associated with the
$\mu$-synthesis problem. By applications of the theory of $\mathbb
E$-contractions, we obtain several results on decompositions of contractions.
