Einstein-Hilbert action with cosmological term from Chern-Simons gravity

We propose a modification to the Lie algebra $S$-expansion method. The modification is carried out by imposing a condition on the $S$-expansion procedure, when the semigroup is given by a cyclic group of even order. The $S$-expanded algebras are called $S_H$-expanded algebras where $S=Z_{2n}$. The invariant tensors for $S_H$-expanded algebras are calculated and the dual formulation of $S_H$-expansion procedure is proposed. We consider the $S_H$-expansion of the five-dimensional $AdS$ algebra and its corresponding invariants tensors are found. Then a Chern-Simons Lagrangian invariant under the five-dimensional $AdS$ algebra $S_H$-expanded is constructed and its relationship to the general relativity is studied.