Definition of trace of an adjoint representation.

Question

What does the trace of an adjoint representation mean? I was asked to prove the following

If $z \in L'$, then $\operatorname{tr}(\operatorname{ad}z)=0$.

I know what a trace of a matrix is, but trace of an adjoint map is not making any sense to me. Note: $L$ is a lie-algebra and $L'=[L,L]$.

Answer 1

If $z\in L$, then $\textrm{ad}\,z$ is a linear map from $L$ to $L$. A linear map from a finite-dimensional vector space to itself has a well-defined trace. Just take the trace of the matrix representing the linear map with respect to any basis of the vector space.

Answer 2

Since $z\in [L,L]$ is a linear combination of elements $[x,y]$ and $$ {\rm tr} ({\rm ad}([x,y]))={\rm tr} ([{\rm ad}(x),{\rm ad}(y)])=0, $$ the claim follows.