Proposition 9

Suppose that $K$ is a field and that the Lie algebra $g$ is finite-dimensional over $K$. *Let $a$ be an ideal of $g$, $\beta$ the Killing form of $g$ and $\beta'$ the Killing form of $a$. Then $\beta'$ is the restriction of $\beta$ to $a$.

Let $u$ be an endomorphism of the vector space $g$ which leaves $a$ stable. Let $v$ be the restriction of $u$ to $a$ and $w$ the endomorphism of the vector space $g/a$ derived from $u$ when passing to the quotient. Then $\operatorname{Tr} u = \operatorname{Tr} v + \operatorname{Tr} w$ as is seen by taking a basis $(x_1, \ldots, x_n)$ of $g$ of which the first $p$ elements form a basis of $a$. Then let $x \in a, y \in a$ and apply the above formula to the case where $u = (\operatorname{ad}_g x)(\operatorname{ad}_g y)$. Then $v = (\operatorname{ad}_a x)(\operatorname{ad}_a y)$ and $w = 0$. Hence $\beta(x, y) = \beta'(x, y)$.