  A classical result of Sz.-Nagy asserts that a Hilbert-space contraction
operator $T$ can be lifted to an isometry $V$. A more general multivariable
setting of recent interest for these ideas is the case where (i) the unit disk
is replaced by a certain domain contained in ${\mathbb C}^3$ (called the {\em
tetrablock}), (ii) the contraction operator $T$ is replaced by a commutative
triple $(T_1, T_2, T)$ of Hilbert-space operators having ${\mathbb E}$ as a
spectral set (a tetrablock contraction) . The rational dilation question for
this setting is whether a tetrablock contraction $(T_1, T_2, T)$ can be lifted
to a tetrablock isometry $(V_1, V_2, V)$ (a commutative operator tuple which
extends to a tetrablock-unitary tuple $(U_1, U_2, U)$---a commutative tuple of
normal operators with joint spectrum contained in the distinguished boundary of
the tetrablock). We discuss necessary conditions for a tetrablock contraction
to have a tetrablock-isometric lift. We present an example of a tetrablock
contraction which does have a tetrablock-isometric lift but violates a
condition previously thought to be necessary for the existence of such a lift.
Thus the question of whether a tetrablock contraction always has a
tetrablock-isometric lift appears to be unresolved at this time.
