Sharp estimnew series of simple Lie algebras of characteristic 3

Three new series of simple finite-dimensional Lie algebras over a field of characteristic 3 are constructed. For explicit realization, these algebras can be represented as a sum of a Lie algebra of general or special Cartan type and of certain of its tensor modules. The algebras have gradings of depth 2 or 4, with a classical zero component; and the author gives a characterization for them in the class of all graded Lie algebras. In order to prove nonisomorphism of these algebras with the Lie algebras of Cartan type, the subalgebras containing a certain invariant set are studied. With a view of systematizing the known examples of simple finite-dimensional Lie algebras of characteristic 3, analogous realizations of two previously known series are also presented.