Remark

(1) Let $K'$ be a commutative ring containing K. If $A(X)$, $L(X)$ and $L_{K'}(X)$ are identified with subsets of $A_{K'}(X)$, we deduce from part (a) of Theorem 1 the relation

(2) Corollary 2 to Theorem 1 remains valid if it only assumes that the additive group of the ring K is torsion-free. For suppose first that $K = \mathbf{Z}$; every primitive element of $A(X)$ is a primitive element of $A_q(X)$ and hence is in $L_q(X) \cap A(X) = L(X)$ (Corollary 2 and formula (1)). In the general case, K is flat over $\mathbf{Z}$ and we apply Remark 2 of § 1, no. 2 and Proposition 3 of § 2, no. 5.

(3) Let $\Delta$ be a commutative monoid, $\phi_0$ a mapping of $X$ into $\Delta$ and $\phi : Mo(X) \to \Delta$ the homomorphism of the associated monoid; if $A(X)$ is given the graduation $(A^\delta(X))_{\delta \in \Delta}$ defined in Algebra, Chapter III, § 3, no. 1, Example 3 and $L(X)$ the graduation $(L^\delta(X))_{\delta \in \Delta}$ defined in § 2, no. 6, we have immediately, for $\delta \in \Delta$, $L^\delta(X) \subset L(X) \cap A^\delta(X)$. As $L$ is the sum of the $L^\delta(X)$ for $\delta \in \Delta$, and the sum of the $L(X) \cap A^\delta(X)$ for $\delta \in \Delta$ is direct, this implies

(4) Let $A$ be a unital associative algebra and $t = (t_i)_{i \in I}$ a family of elements of $A$. We have a diagram

where $i$ is the canonical injection, $f_t$ is the Lie algebra homomorphism defined by $t$ and $g_t$ is the unital algebra homomorphism such that $g_t(i) = t_i$ for $i \in I$. The diagram is commutative for $g_t \circ i$ and $f_t$ coincide on $I$. It follows that if $P \in L(I)$, the element $P((t_i)_{i \in I})$ defined in § 2, no. 4, coincides with the element $P((t_i)_{i \in I})$ defined in Algebra, Chapter III, § 2, no. 8, Example 2.