Remark

Suppose that K is a field of characteristic zero and let Q be the projector of $A(X) = U(L(X))$ onto $L(X)$ associated with the canonical graduation of $U(L(X))$, cf. § 1, no. 5. For $\alpha \in \mathbf{N}^{(X)}$, $Q(A^\alpha(X)) \subset L^\alpha(X)$. For it suffices to verify that the image and the kernel of Q are graded submodules of $A(X)$ with the graduation of type $\mathbf{N}^{(X)}$. This is obvious for the image, which is equal to $L(X)$. On the other hand, let $n$ be an integer $\geq 1$. The vector subspace of $A(X)$ generated by the $y^n$, where $y \in L(X)$, is equal to the vector subspace of $A(X)$ generated by the $\sum_{\sigma \in S_n} y_{\sigma(1)} y_{\sigma(2)} \cdots y_{\sigma(n)}$, where $y_1, y_2, \ldots, y_n$ are homogeneous elements of $L(X)$; then this subspace is a graded submodule of $A(X)$.

(Note that, if Card(X) $\geq 2$, the projectors P and Q do not coincide on $A^+(X)$. For let $x, y$ be in X with $x \neq y$ and write

Then $Q(z) = 0$ and $P(z) = \frac{1}{3}[x, [x, y]] \neq 0$, cf. § 2, no. 10, Example and no. 11, Theorem 1.)