In Hitchin's paper Lie group and Teichmuller space,

http://www.sciencedirect.com/science/article/pii/004093839290044Ihe mentioned in section 4 that a regular nilpotent element $e$ is an element that is nilpotent with respect to adjoint action and its dimension of centralizer is equal to the rank $l$ of the complex simple lie algebra $\mathfrak{g}^{c}$.

But I also read from the construction of Cartan subalgebra, a regular element $e$ whose dimension of centralizer is minimized, i.e. equal to the rank of lie algebra $l$ must lie in the Cartan subalgebra $\mathfrak{h}=\{v\in\mathfrak{g}^{c} | (ad_e)^kv=0, k\in \mathbb{N}\}$ it spans. In particular, it is semisimple. My confusion is how can such an element be both semisimple and nilpotent with respect to adjoint action? Could someone remind me what I mess up here?