Relations of derivations.

LEMMA 5.5   Let $A$ be a ring. Let $B$ be a commutative $A$ -algebra. For any $x \in \operatorname{Der}_A(B)$ , we have

$$[L_x, d]= 0$$

PROOF.. This is based on the fact that $d$ is naturally defined. A direct proof is obtained by using Lemma 5.4 and by noting the following facts.

1. $$(L_x d) f= L_x (df )=d (x. f)= d L_x f$$

2. $$(L_x d) d g =0 , \qquad (d L_x) d g =d (d L_x(g))=0$$

$\qedsymbol$

LEMMA 5.6   Let $A$ be a commutative ring. Let $B$ be a commutative $B$ -algebra.

1. For any $x \in \operatorname{Der}_A(B)$ , we have
   $$[d, i_x]_+ =L_x.$$

2. For any $x,y \in \operatorname{Der}_A(B)$ , we have
   $$[L_x,i_y]=i_{[x,y]}.$$

PROOF.. Same method as above works. $\qedsymbol$