  The purpose of this paper is to investigate $(n+1)$-Lie algebras induced by
$n$-Lie algebras and trace maps. We highlight a comparison of their structure
properties (solvability, nilpotency) and the cohomology groups as well as
central extensions. Moreover, we provide for dimensions $n$, $n+1$ and $n+2$,
the classification of $n$-Lie algebras which are induced by $(n-1)$-Lie
algebras.
