  By applying the stress-tensor-scalar operator product expansion (OPE) twice,
we search for algebraic structures in $d=4$ conformal field theories (CFTs)
with a pure Einstein gravity dual. We find that a rescaled mode operator
defined by an integral of the stress tensor $T^{++}$ on a $d=2$ plane satisfies
a Virasoro-like algebra when the dimension of the scalar is large. The
structure is enhanced to include a Kac-Moody-type algebra if we incorporate the
$T^{--}$ component. In our scheme, the central terms are finite. It remains
challenging to directly compute the stress-tensor sector of $d=4$ scalar
four-point functions at large central charge, which, based on holography and
bootstrap methods, were recently shown to have a Virasoro/${\cal W}$-algebra
vacuum block-like structure.
