  Any graded restricted simple Lie algebra of Cartan type contains a subalgebra
isomorphic to the Witt algebra over a field of prime characteristic. As some
analogue of study on branching rules for restricted non-classical Lie algebras,
it is shown that each graded restricted simple Lie algebra of Cartan type can
be decomposed into a direct sum of restricted baby Verma modules and simple
modules as an adjoint module over the Witt algebra. In particular, the
composition factors are precisely determined.
