Diagonalization of the Toda flow on conjugacy classes of matrices with simple spectrum

We construct coordinates on conjugacy classes of traceless complex matrices with simple spectrum that diagonalize the non-periodic Toda vector field. By this we mean that the coordinates, defined on an open and dense neighborhood of any diagonal matrix in the conjugacy class, decouple the Toda vector field into a sum of multiples of the Euler and rotational vector field in $\mathbb{C}$. Using Lie theoretic methods, we extend this construction from ${\mathfrak{s}\mathfrak{l}}_{\mathbb{C}}$ to arbitrary complex semisimple Lie algebras and to their real forms