Prove that 1. $\kappa(x,y)$ is a symmetric bilinear form? 2. $\kappa([x,y],z)=\kappa(x,[y,z])$

Question

Let $\mathfrak g$ be a complex Lie algebra. The Killing form on $\mathfrak g$ is defined by $\kappa(x,y)=tr(ad(x) \circ ad(y))$.

Now how to prove that

1. $\kappa(x,y)$ is a symmetric bilinear form?
2. $\kappa([x,y],z)=\kappa(x,[y,z])$

I am actually in between the trace and lie algebra cases. I have proved that $ad(X)Y=[X,Y]$. So do I have to calculate it on to the elementary matrices?

I am not getting any clue. Please help.

Answer 1

$tr(AB)=tr(BA)$ implies 1.

$ad([x,y])=ad(x)ad(y)-ad(y)ad(x)$ Jacobi identity.

$tr(ad([x,y])ad(z))=tr((ad(x)ad(y)-ad(y)ad(x))ad(z))$

$tr(adx(ad([y,z])=tr(ad(x)(ad(y)ad(z)-ad(z)ad(y))$

use $tr(AB)=tr(BA)$ which implies that $tr(ad(y)(ad(x)ad(z))=tr(ad(y)(ad(x)ad(z))=tr((ad(x)ad(z))ad(y))$.