  A proof of Poincar\'e-Birkhoff-Witt theorem is given for a class of
generalized Lie algebras closely related to the Gurevich S-Lie algebras. As
concrete examples, we construct the positive (negative) parts of the quantized
universal enveloping algebras of type A_n and M_{p,q,e}(n,K), which is a
non-standard quantum deformation of GL(n). In particular, we get, for both
algebras, a unified proof of the Poincar\'e-Birkhoff-Witt theorem and we show
that they are genuine universal enveloping algebras of certain generalized Lie
algebras.
