  We establish some results on the structure of the geometric unipotent
radicals of pseudo-reductive k-groups. In particular, our main theorem gives
bounds on the nilpotency class of geometric unipotent radicals of standard
pseudo-reductive groups, which are sharp in many cases. A major part of the
proof rests upon consideration of the following situation: let k' be a purely
inseparable field extension of k of degree p^e and let G denote the Weil
restriction of scalars R_{k'/k}(G') of a reductive k'-group G'. When G=
\R_{k'/k}(G') we also provide some results on the orders of elements of the
unipotent radical \RR_u(G_{\bar k}) of the extension of scalars of G to the
algebraic closure \bar k of k.
