Unipotent element of linear algebraic group

Definition

Finite-dimensional case

Suppose $G$ is a linear algebraic group over a field $k$, i.e., $G$ is identified as a closed subgroup of the general linear group $GL(n,k)$. Denote the identity element of $G$ by $1$.

An element $g\in G$ is termed unipotent if $g-1$ (the subtraction being done with respect to the additive structure of matrices) is a nilpotent element in the matrix ring $M(n,k)$. By degree considerations, this is equivalent to requiring that $(g-1)^{n}=0$, where the powering is done in the matrix ring.