Why is derivation $D$ on bilinear form $C:V\times V\to K$ is defined as $C(DX,Y)+C(X,DY)=0$?

Question

This is related to Knapp, Lie groups, Chpt I, Sec 4, Examples 2).

Let $V$ be a vector space over $k$ and $C$ be a bilinear form over $V\times V$. Derivation $D$ on bilinear form $C:V\times V\to K$ is defined as $C(DX,Y)+C(X,DY)=0$ for all $X,Y\in V$.

$\textbf{Q:}$ Why do I want $C(DX,Y)+C(X,DY)=0$?

Answer 1

The condition means that the bilinear form is derivation invariant, see this paper, and this terminology makes more sense. For Lie algebras, it corresponds to the fact that the Killing form is invariant. It is given by $$C(x,y)=tr({\rm ad}(x),{\rm ad}(y))$$ for $x,y\in V$.