Proposition 8

Let $A$ be a graded K-algebra.

(i) For every $\varepsilon$-derivation $d : A \to A$ and every homogeneous element $a$ of $A$,

(ii) If the algebra $A$ is associative, $\mathrm{ad}_\varepsilon(a)$ is an $\varepsilon$-derivation of $A$ of degree $\deg(a)$.

(i) Suppose that $d$ is of degree $\delta$, let $\alpha = \deg a$ and write $f = [d, \mathrm{ad}_\varepsilon(a)]_\varepsilon$. For every homogeneous element $x \in A$ of degree $\xi$, we have, by (1) (no. 1),

(ii) For all $x$ homogeneous of degree $\xi$ and all $y$ homogeneous of degree $\eta$ in $A$,

taking account of (1) and the associativity of $A$.

When $A$ is associative, $\mathrm{ad}_\varepsilon(a)$ is called the inner $\varepsilon$-derivation of $A$ defined by $a$.