In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.”
A unipotent element $g$ has $g_s=1$, and $1$ is not regular. Hence there are no regular unipotent elements according to Borel.
However, “regular unipotent elements” are well studied under the definition “An element $x$ of a linear algebraic group $G$ is called regular if $\dim(C_G(x))$ is smallest possible among all elements of $G$.” as in Malle–Testerman page 115.
- Can people confirm that these definitions contradict each other?
- Is the Malle–Testerman definition more common now?
Humphrey's Linear Algebraic Groups carefully avoids talking about regular elements, and only discusses regular semisimple elements where the two definitions agree.