Introduction. We study a family of Lie algebras of characteristic p which are defined as subalgebras of the derivation algebra of the group algebra of an elementary p-group. In particular we show that simple Lie algebras of dimensions m (pn -1), mpn, pn_ 2, where m and n are arbitrary integers such that 1 2 only for the dimensions pn and pn 2, are associated with this family. The algebras studied by M. S. Frank [2] are included in our family, but those of dimension m(pn -1) in general appear to be new. Since this paper was written, the paper of A. A. Albert and M. S. Frank