Jordan decompositions

If F is an algebraically closed field, any element in M n (F) is similar to a sum of a diagonal matrix and a nilpotent matrix whose non-zero entries are all 1, just above the diagonal. Something similar is true for elements of an arbitrary affine algebraic group as well as its Lie algebra. That's what this essay will attempt to explain. I begin with a very elementary account of what happens for M n , then go on to use what might be called Tannakian methods to deal with the general case. My main references, in addition to the items by Newton, have been [Serre:1965] and [Springer:1998]. Section 3 was suggested by §2.5 of Springer's book.