  We show that for four-dimensional spacetimes with a non-null hypersurface
orthogonal Killing vector and for a Chern-Simons (CS) background
(non-dynamical) scalar field, which is constant along the Killing vector, the
source-free equations of CS modified gravity decouple into their Einstein and
Cotton constituents. Thus, the model supports only general relativity
solutions. We also show that, when the cosmological constant vanishes and the
gradient of the CS scalar field is parallel to the non-null hypersurface
orthogonal Killing vector of constant length, CS modified gravity reduces to
topologically massive gravity in three dimensions. Meanwhile, with the
cosmological constant such a reduction requires an appropriate source term for
CS modified gravity.
