Find the highest weight and highest weight vector for adjoint representation of $\mathfrak{sl}(n)$

Question

By definition, the highest weight vector for $\mathfrak{sl}(n)$ is an eigenvector for the action of a Cartan subalgebra which is killed by the adjoint action of $E_{ij}$ with $i < j$ (which correspond to the eigenspaces of positive roots). Let us choose, as a Cartan subalgebra, the subalgebra of all diagonal matrices of $\mathfrak{sl}(n)$. Now, the highest weight vector must be of type $E_{kh}$ with $k \neq h$ because all eigenvectors of non zero eigenvalue are elementary matrices of this type. What I find most difficult now is to find an elementary matrix $E_{kh}$ such that $[E_{ij}, E_{kh}] = 0$ for every $i < j$, because this happens everytime $j \neq k$ and $h \neq i$. What am I missing here? Is this even the right way to go about this problem?

Answer 1

Take $E_{1n}$. Then, whenever $i<j$, $[E_{ij},E_{1n}]=0$.