Unipotent element of linear algebraic group

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.