I'm not sure where your "statement" comes from, or why the specific type of embedding of $H$ is assumed here. But the closest relative of this situation I'm aware of goes back to work of Kostant (in characteristic 0) and later Steinberg (more generally), in which they consider a sort of adjoint quotient of a semisimple (or reductive) group. Only the semisimple classes of such a group are closed, so one has to settle for something less than a strict "quotient" construction.

In your concrete situation, the significance of assigning to a matrix the nontrivial coefficients of its characteristic polynomial originates with the behavior of semisimple elements under the Weyl group action: affine space of dimension $r$ provides a model of the orbit space $T/W$ when $\dim T = r$ (the rank of $H$ when $T$ is a maximal torus). It's probably useful to consult $\S6$ in Steinberg's paper on regular elements here. In my 1995 AMS monograph on conjugacy classes in semisimple algebraic groups, I wrote up some of the relevant material in Chapter 3.

For an arbitrary matrix in $H$, the only information given by its characteristic polynomial is the list of eigenvalues with multiplicity. Since the semisimple elements of $H$ are Zariski-dense, their conjugacy classes are effectively parametrized in this way, and by the standard structure theorems this boils down to the orbit space $T/W$. Technically it's all a bit complicated, and in an abstract formulation (say for simply connected groups) one works with the values of fundamental representations rather than the explicit characteristic polynomials here.

I'm not sure where your "statement" comes from, or why the specific type of embedding of $H$ is assumed here. But the closest relative of this situation I'm aware of goes back to work of Kostant (in characteristic 0) and later Steinberg (more generally), in which they consider a sort of adjoint quotient of a semisimple (or reductive) group. Only the semisimple classes of such a group are closed, so one has to settle for something less than a strict "quotient" construction.

In your concrete situation, the significance of assigning to a matrix the nontrivial coefficients of its characteristic polynomial originates with the behavior of semisimple elements under the Weyl group action: affine space of dimension $r$ provides a model of the orbit space $T/W$ when $\dim T = r$ (the rank of $H$ when $T$ is a maximal torus). It's probably useful to consult $\S6$ in Steinberg's paper on regular elements here. In my 1995 AMS monograph on conjugacy classes in semisimple algebraic groups, I wrote up some of the relevant material in Chapter 3.

For an arbitrary matrix in $H$, the only information given by its characteristic polynomial is the list of eigenvalues with multiplicity. The semisimple elements of $H$ are Zariski-dense, and their conjugacy classes are effectively parametrized in this way. By the standard structure theorems this boils down to the orbit space $T/W$. Technically it's all a bit complicated, and in an abstract formulation (say for simply connected groups) one works with the values of fundamental representations rather than the explicit characteristic polynomials here.