  This paper provides new, relatively simple proofs of some important results
about unipotent classes in simple linear algebraic groups. We derive the
formula for the Jordan blocks of the Richardson class of a parabolic subgroup
of a classical group. This result was originally due to Spaltenstein. Secondly,
we derive, for good characteristic, the description of the natural partial
order of unipotent classes of a classical group in terms of their Jordan
blocks. This result was originally due to Gerstenhaber and Hesselink. As a
consequence we obtain a proof of the Bala--Carter Theorem which holds even in
certain bad characteristics (this proof requires the prior classification of
unipotent classes, unlike the original proofs due to Bala, Carter and
Pommerening).
