Example

Propositions 9 and 10 apply in particular to the graded bigebra $S(M)$ and the skew graded bigebra $\Lambda(M)$. The inner products by elements of degree 1 in $S(M)$ (resp. $S(M)^{*gr}$) are derivations which commute with one another, since $S(M)$ (resp. $S(M)^{*gr}$) is commutative.

Similarly, the inner products by elements of degree 1 in $\Lambda(M)$ (resp. $\Lambda(M)^{*gr}$) are antiderivations, which are of zero square, for the square of an element of degree 1 in the algebra $\Lambda(M)$ (resp. $\Lambda(M)^{*gr}$) is zero.