  In the representation theory of simple Lie algebras, we consider the problem
of constructing a "canonical" weight basis in an arbitrary irreducible
finite-dimensional highest weight module. Vinberg suggested a method for
constructing such bases by applying the lowering operators corresponding to all
positive roots to the highest weight vector. He proposed several conjectures on
the parametrization and structure of such bases. It is already known that these
conjectures are true for the cases $A_n$, $C_n$, $G_2$, $B_3$. In this paper,
we prove these conjectures for the Lie algebra of type $D_4$.
