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Some time ago, the standard geometric framework of Einstein gravity wasextended by gauging the Maxwell algebra as well as the so called AdS-Maxwellalgebra. In this letter it is shown that the actions for these four-dimensionalextended Einstein gravities can be obtained from the five-dimensionalChern-Simons gravities actions by using the Randall-Sundrum compactificationprocedure. It is found that the In\"on\"u-Wigner contraction procedure, in theWeimar-Woods sense, can be used both to obtain the Maxwell-Chern-Simons actionfrom the AdS-Maxwell-Chern-Simons action and to obtain the Maxwell extension ofEinstein gravity in 4D from the four-dimensional extendedAdS-Maxwell-Einstein-Hilbert action. It is also shown that the extendedfour-dimensional gravities belongs to the Horndeski family of scalar-tensortheories.

Einstein gravity with generalized cosmological term from five-dimensional AdS-Maxwell-Chern-Simons gravity