I am trying to understand this statement on p.208 of Varadarajan's book Lie Groups, Lie Algebras, and Their Representations. Let $\mathfrak{g}$ be a Lie algebra over a field $k$ of characteristic 0. There is a statement that for any derivation $D$ of $\mathfrak{g}$, $\langle DX, Y \rangle + \langle X,DY \rangle =0$.
He says that it follows from a certain fact. If det$(T - \text{ad} X)$ is written as $\sum_{0 \leq i \leq m}(-1)^{m-1}p_i(X)T^i$, then if $\overline{D}$ is defined on page 207, $\overline{D} p_i=0$ for all $i$. More specifically, if $D$ is an endomorphism of the underlying vector space of $\mathfrak{g}$, then $\overline{D}$ is defined to be the derivation of the algebra of polynomial functions from $\mathfrak{g}$ to $k$ such that if $\lambda: \mathfrak{g} \rightarrow k$ is a linear function $(\overline{D} \lambda)(X)=-\lambda (DX)$ for $X \in \mathfrak{g}$.
How does the property of the Killing form follow from that fact?
