[en] F-Lie algebras are natural generalizations of Lie algebras (F=1) and Lie superalgebras (F=2). When F>2 not many finite-dimensional examples are known. In this article we construct finite-dimensional F-Lie algebras F>2 by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realizations of F-Lie algebras constructed in this way from su(n),sp(2n) so(n) and sl(n|m), osp(2|m) are given. We obtain nontrivial extensions of the Poincare algebra by Inoenue-Wigner contraction of certain F-Lie algebras with F>2