Proving that a symmetric bilinear form on $L$ is invariant

Question

Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$

Where $\text{ tr }$ denotes the trace of endomorphisms.

$(a)$ Prove that this form is invariant, i.e., we have $$\langle [x,y], z \rangle_{\rho} = \langle x,[y,z] \rangle_{\rho}$$

for all $x,y,z \in L.$

My thoughts:

I do not know how exactly to do this, specifically how to expand the trace. I know that the Lie bracket of a Lie Algebra satisfies antisymmetry and Jacobi identity but still how to use this to prove the required. Could anyone help me in this please?

Answer 1

Any trace form is invariant because of ${\rm tr}([A,B]C)={\rm tr}(A[B,C])$ for $A,B,C \in {\rm End}(V)$.

Indeed, using Jyrki's comment with ${\rm tr}(BAC)={\rm tr}(ACB) $ we have $$ {\rm tr}([A,B]C)={\rm tr}((AB-BA)C)={\rm tr}(ABC-BAC)={\rm tr}(ABC-ACB)={\rm tr}(A[B,C]). $$