 Units of a finite-dimensional algebra

The unit group $E^{\times }$ is a nonempty open subset of the affine space $E$, detected by the determinant of left multiplication. It is therefore an irreducible connected linear algebraic group. Its map to $(E/J(E){)}^{\times }$ has kernel $1+J(E)$, a closed unipotent subgroup since the radical is nilpotent.