Corollary

Suppose that $\Delta = \mathbf{Z}$. Then: (i) The square of an antiderivation is a derivation. (ii) The bracket of two derivations is a derivation. (iii) The bracket of an antiderivation and a derivation of even degree is an antiderivation. (iv) If $d_1$ and $d_2$ are antiderivations, $d_1 d_2 + d_2 d_1$ is a derivation.

Under the hypotheses of the beginning of this no., consider now a finite sequence $D = (d_i)_{1 \leq i \leq n}$ of pairwise permutable derivations of $(A, A', A'')$. For every polynomial $P(X_1, \ldots, X_n)$ in the algebra $K[X_1, \ldots, X_n]$, the element $P(d_1, \ldots, d_n)$ of $\mathrm{Endgr}_K(A \oplus A' \oplus A'')$ is then defined (§ 2, no. 9); its abbreviated notation is $P(D)$.