I see the subgroup $$\left\{\begin{pmatrix}1 & x \\ 0 & 1\end{pmatrix}\right\} \subset \mathrm{GL}(2)$$ written as $U$ and described as the unipotent subgroup of $\mathrm{GL}(2)$ in some literature, and this makes sense to me. I sometimes see it written as $N$ and described as the standard nilpotent subgroup of $\mathrm{GL}(2)$ in other literature though (maybe only for finite fields?). Similar for $\mathrm{GL}(n)$.
Which is more common or more correct?
Let U = {U_x = (1 x; 0 1)} be a Unipotent group in GL2 Then we have the central series: C^1(U) = U C^2(U) = [U,U]= {(1 0 ; 0 1)} which says that the group U is nilpotent.
On the other hand, if N = {N_x=(0 x; 0 0)}; which consists of nilpotent matrices since N^2=0. Then C^1(U_x) = U_x = I + N_x and C^2(U_x) = I + (N_x)^2 = I. Hence,the descending central series C^2(U) = I.
This can be generalised to GL(n).