  The Einstein-Hilbert action with a cosmological constant is the most general
local four-dimensional action leading to second-order derivative equations of
motion that are symmetric and divergence free. In higher dimensions, additional
terms can appear. We investigate a generalised metric decomposition involving a
scalar degree of freedom to express the higher-dimensional action as an
effective four-dimensional scalar-tensor theory. From the higher-dimensional
Ricci scalar alone and a subclass of our metric ansatz, we recover the subset
of Horndeski theories with luminal speed of gravitational waves. More
generally, beyond-Horndeski terms appear. When including a Gauss-Bonnet scalar
in the higher-dimensional action, we generate contributions to all cubic-order
second-derivative terms present in the degenerate higher-order scalar-tensor
theory as well as higher-derivative terms beyond that. We discuss this
technique as a way to generate healthy four-dimensional gravity theories with
an extra scalar degree of freedom and outline further generalisations of our
method.
