I'm currently trying to learn about regular elements of a Lie algebra but i'm finding the definition quite abstract and can't seem to find many examples anywhere.
One thing i'm really unsure about is that I have read that in $\mathfrak{sl}(3)$ a regular element is $X= E_{1,2} + E_{2,3}$ where $E_{x,y}$ corresponds to the matrix with a 1 in position (x,y) and 0's elsewhere. The centraliser of this group is given by $\text{span} \{E_{1,3} \}$. A theorem then claims that this set, which is equal to $\mathfrak{g}^0(X)$ is a Cartan subalgebra. But I on't see how this is true?
I was wondering if someone could give me an example of how to determine the regular elements of a given Lie algebra, say $\mathfrak{sl}(3)$.
There is apparently more than one definition of "regular element".
The definition that relates to Cartan subalgebras is: $x \in \mathfrak{g}$ is regular if $$\mathfrak{g}_0(x) = \{y \in \mathfrak{g}: \; \mathrm{ad}(x)^n(y) = 0 \; \mathrm{for} \; \mathrm{some} \; n\}$$ has minimal dimension among all elements of $\mathfrak{g}.$ In this case, $\mathfrak{g}_0(x)$ is a Cartan subalgebra. This is not the centralizer.
Your matrix $$X = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$$ does not satisfy this criterion. Even though its centralizer is $2$-dimensional (which looks like the right number), the kernel of $\mathrm{ad}(X)^2$ is larger ($5$-dimensional).
In fact the regular elements of $\mathfrak{sl}(n,\mathbb{C})$ are exactly the matrices with no repeated eigenvalues.