The definition of the Killing form

Question

We know that the definition of the Killing form: $\kappa:L\times L\rightarrow F$ with $\kappa\left(x,y\right)=\rm{Tr}\left(ad(x)\cdot ad(y)\right)$.

Then we have a property of the Killing form:$\kappa\left(\left[x,y\right],z\right)=-\kappa\left(y,\left[x,z\right]\right)$.

How can I verify that if we define the Killing form to be: A quadratic form $\kappa:L\times L\rightarrow F$ which satisfies $\kappa\left(\left[x,y\right],z\right)=-\kappa\left(y,\left[x,z\right]\right)$ ,then $\kappa\left(x,y\right)$ must be $\rm{Tr}\left(ad(x)\cdot ad(y)\right)$.

Thanks in advance.

Answer 1

This need not be true. Consider the trace form $T:L\times L\to F$, assuming $L$ to be a Lie algebra of matrices, given by $(x,y)\mapsto tr(xy)$. It is quadratic and satifies the invariance condition, but need not coincide with the Killing form. For simple Lie algebras over characteristic zero the trace form is a non-zero multiple of the Killing form (the multiple need not be $1$, consider $\mathfrak{sl}(n)$); for nilpotent Lie algebras the Killing form is identically zero, but the trace form may be different from zero (take the abelian Lie algebra of diagonal matrices).