Constructing (n+ 1)-Lie algebras fromn-Lie algebras

n -Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we construct ( n + 1)-Lie algebras by two-dimensional extensions of metric n -Lie algebras, and construct ( n + 1)-Lie algebras in terms of n -Lie algebras and certain linear functions. Along this approach, we obtain ( n + 1)-Lie algebras by n -Lie algebras with the generating index n , and 4-Lie algebras by non-nilpotent 3-Lie algebras with the generating index 4.