Proposition 10

Suppose that $K$ is a field and that the Lie algebra $g$ is finite-dimensional over $K$. The Killing form $\beta$ of $g$ is completely invariant.

Let $D$ be a derivation of $g$. There exists a Lie algebra $g'$ containing $g$ as an ideal of codimension 1 and an element $x_0$ of $g'$ such that $Dx = [x_0, x]$ for all $x \in g$ (§ 1, no. 8, Example 1). Let $\beta'$ be the Killing form of $g'$. For $x, y$ in $g$, $\beta'([x, x_0], y) = \beta'(x, [x_0, y])$, that is $\beta'(Dx, y) + \beta'(x, Dy) = 0$. Now the restriction of $\beta'$ to $g$ is $\beta$ (Proposition 9). Hence the proposition.