The last years brought a new technique of constructing simple Lie algebras of small characteristic \(p\), \((p=2,3,5)\). Such algebras are a) isomorphic to neither classical nor Cartan type Lie algebras; b) closely related to both the first and the second ones; c) direct sum of some Cartan type Lie algebra and their modules, and the multiplication is given by invariant differential operators; d) provided with natural grading the nonpositive part which coincides with the nonpositive part of some standard grading of classical Lie algebra.
As a result of such symbiosis three new series of algebras are constructed in the paper. These series are \(D_4(3:{\mathbf m})\), \(G_2(2:{\mathbf m})\), \(C_{3,a}(2:{\mathbf n},1)\). The last one is related to the Kac-Weisfeiler Lie algebra associated with the Cartan matrix \(\left(\begin{smallmatrix} 0 & 1 & 0 \\ a & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}\right)\) instead of a classical one. It is proved that these algebras are not isomorphic to known simple Lie algebras.