I want to understand (better) the definition and meaning of regular elements of semisimple Lie groups resp. Lie algebras.
A regular element $X$ of $\mathfrak g$ is one whose centralizer has smallest dimension possible. I admit, that I have no intuition for this condition. Considering the characteristic polynomial of $Ad(X)$, let $D_j(X)$ be its coefficients and let $r$ be the smallest number such that $D_r$ is not identically the zero function, then $X$ is regular iff $D_r(X) \neq 0$.
Example: Let $\mathfrak g = sl(2,\mathbb R)$ with basis $h,e,f$ satisfying $[h,e]=2e, [h,f]=-2f, [e,f]=h$. If $w = ae + bf + ch = \begin {pmatrix} c&a \\ b & -c\end {pmatrix}$, then I compute $ad (w) = \begin{pmatrix} 0 & -b & a\\ -2a & 2c & 0\\ 2b & 0 & -2c\end{pmatrix}$ with respect to above basis. Therefore I get for the characteristic polynomial of ad w: $\lambda^3 + \lambda(-4c^2-4ab)$. Is my computation correct? Thus by definition, $w$ is regular iff $c^2+ab \neq 0$.
Now I read that for $\mathfrak g = sl(n,\mathbb C)$ the regular elements on the Lie group level are the matrices with $n$ distinct eigenvalues. Why is that? E.g.,why does the above condition for regularity on the Lie algebra translate to the condition of distinct eigenvalues on the Lie group?
For $G=SL(2,\mathbb R)$ I know there are two classes of Cartan subgroups: the elliptic elements which exponentiate from the Cartan subalgebra (CSA) $\mathbb R h$ and then the hyperbolic elements which exponentiate from the CSA $\mathbb R (e-f)$. The non-regular elements are the so-called parabolic elements, which have either repeated eigenvalue +1 or repeated eigenvalue -1. The condition of regularity can be expressed by $|tr g| \neq 2$ for $g \in G$.
I would like to understand how this example generalizes to other semisimple groups. The next examples I would like to understand are $SL(n,\mathbb R)$ and $Sp(2,\mathbb R)$. Are the regular elements also the ones with distinct eigenvalues? Can regularity on the Lie group also be expressed in a similar fashion (like above with the trace?).
For $Sp(2,\mathbb R)$ I've read there are four classes of nonconjugate ($\theta$-stable) CSA's. Do they correspond to sth. similar like the classification of elements of $SL(2,\mathbb R)$ by eccentricity?
Please excuse the confusion of my questions which is due to my missing understanding..