How to show that set of diagonal matrices is the maximal toral subalgebra of $sl(n)$

Question

$sl(n)$ is the set of $n\times n$ matrices with trace=0. I know that $sl(n)$ is a finite dimensional simple lie algebra and the maximal toral subalgebra of a finite dimensional semi simple lie algebra is abelian. The set of diagonal matrices is abelian.

I mean I don't know how to go around this problem.

Answer 1

A torus $\mathfrak h$ in a simple Lie algebra $L$ is maximal if and only if it is self centralizing: $\mathfrak h = C_L(\mathfrak h)$.

So prove that if a matrix $M \in \mathfrak{sl}_n$ commutes with all diagonal matrices in $\mathfrak{sl}_n$ then $M$ must be diagonal.