A remark on the space of invariant differential operators

We prove that the problem of describing differential operators which are invariant with respect to a Gartan type Lie algebra $L=\bigoplus\limits_{i\ge -2}L_i$, can be reduced to describing the spaces of multilinear maps which are invariant with respect to the subalgebra $\mathscr{L}_+=\bigoplus\limits_{i>0}L_i$. These maps take value in the trivial $\mathscr{L}_+$-module.