Unipotent subgroups and nilpotent subgroups

Question

Let $G$ be a connected complex Lie subgroup of $GL_n(\mathbb C)$. Let $N$ be the maximal nilpotent normal subgroup of $G$. Is the center $Z$ of $N$ unipotent?

Answer 1

No, consider the subgroup of diagonal matrices, it is abelian and is not unipotent.

Another example consider the space of $3\times 3$ matrices of the form:

$\pmatrix{a&0&0\cr 0&1&c\cr 0&0&1}$

$\pmatrix{a&0&0\cr 0&1&0\cr 0&0&1}$ is in the center.