Killing form on ${\rm ad}\ L$

Question

What is Killing form on ${\rm ad}\ L$ ?

Note that $L$ has a Killing : $$ \kappa(x,y) = {\rm tr}\ ({\rm ad}_x{\rm ad}_y) $$

Answer 1

The Lie algebra $ad (L)$ is a subalgebra of the general linear Lie algebra $\mathfrak{gl}(n,K)$, which has Killing form $\kappa(X,Y)=2n\; {\rm tr}(XY)-2\, {\rm tr}(X) {\rm tr}(Y)$ for matrices $X,Y$. For example, if $ad (L)\subset \mathfrak{sl}(n,K)$, then $\kappa(X,Y)=2n\; {\rm tr}(XY)$.