Unipotent linear algebraic group

Definition
A linear algebraic group $$G$$ over a field $$k$$ (which therefore comes equipped with an embedding as a closed subgroup of the general linear group $$GL(n,k)$$) is termed a unipotent linear algebraic group if it satisfies the following equivalent conditions:


 * 1) Every element in $$G$$ is a unipotent element, i.e., subtracting $$1$$ from it gives a nilpotent element in the matrix ring.
 * 2) $$G$$ is conjugate in $$GL(n,k)$$ to a subgroup of the upper-triangular unipotent matrix group.

Relation with being nilpotent
Any unipotent linear algebraic group is nilpotent, with its nilpotency class at most $$n - 1$$, where $$n$$ is the degree of the general linear group it is embedded in. However, the converse is not true: it is possible to have a linear algebraic group that is nilpotent as an abstract group but is not unipotent. For instance, the [multiplicative group of a field $$k$$, which is the full group $$GL(1,k)$$, is abelian and hence nilpotent, but is not unipotent.

A related fact in the context of Lie algebras is Engel's theorem.