functoriality of Killing forms

LEMMA 5.27   Let $L$ be a Lie algebra over a field $k$ . Then the followings are true.

1. Let $V$ be a finite dimensional representation of $L$ . Let $W$ be a subrepresentation of $V$ . Then we have
   $$\operatorname{Tr}_V(x y)= \operatorname{Tr}_W (x y)+\operatorname{Tr}_{V/W} (x y).$$

2. Let $I$ be an ideal of $L$ . Assume $L$ is finite dimensional. Then we have
   $$\kappa_L(x, y)=\operatorname{Tr}_{\operatorname{ad}, L}(x y) =\op... ...) =\operatorname{Tr}_{\operatorname{ad},I}(x y)+\kappa_{L/I}(\bar{x}, \bar{y})$$
   (where $\bar{\bullet}$ denotes the class of $\bullet$ in $L/I$ .) In particular, for any $x,y\in I$ , we have
   $$\kappa_L(x, y)= \kappa_I(x, y)$$

PROOF.. (2): We choose a basis $B=B_1\coprod B_2$ of $L$ such that $B_2$ forms a basis of $I$ . Then $\bar{B_1}$ forms a basis of $L/I$ . Under the basis $B$ , $\operatorname{ad}(x)$ may be represented by a matrix

$$\operatorname{ad}_L (x)= \begin{pmatrix} \operatorname{ad}_{L/I}(\bar{x}) & * \\ 0 & \operatorname{ad}_I{x} \end{pmatrix}.$$

We obtain the result easily from this.

(1): may be proved in a same manner. $\qedsymbol$