We set so that is the direct sum of these subspaces, each of which is fixed by .
On the subspace , has the characteristic polynomial . What we want is a single polynomial such that
That is, has no constant term, and for each there is some such that
Thus, if we evaluate on the block we get .
To do this, we will make use of a result that usually comes up in number theory called the Chinese remainder theorem. Unfortunately, I didn’t have the foresight to cover number theory before Lie algebras, so I’ll just give the statement: any system of congruences — like the one above — where the moduli are relatively prime — as they are above, unless is an eigenvalue in which case just leave out the last congruence since we don’t need it — has a common solution, which is unique modulo the product of the separate moduli. For example, the system
has the solution , which is unique modulo . This is pretty straightforward to understand for integers, but it works as stated over any principal ideal domain — like — and, suitably generalized, over any commutative ring.
So anyway, such a exists, and it’s the we need to get the semisimple part of . Indeed, on any block differs from by stripping off any off-diagonal elements. Then we can just set and find . Any two polynomials in must commute — indeed we can simply calculate
Finally, if then so must any polynomial in , so the last assertion of the decomposition holds.
The only thing left is the uniqueness of the decomposition. Let’s say that is a different decomposition into a semisimple and a nilpotent part which commute with each other. Then we have , and all four of these endomorphisms commute with each other. But the left-hand side is semisimple — diagonalizable — but the right hand side is nilpotent, which means its only possible eigenvalue is zero. Thus and .
We recall that any linear endomorphism of a finite-dimensional vector space over an algebraically closed field can be put into Jordan normal form: we can find a basis such that its matrix is the sum of blocks that look like
where is some eigenvalue of the transformation. We want a slightly more abstract version of this, and it hinges on the idea that matrices in Jordan normal form have an obvious diagonal part, and a bunch of entries just above the diagonal. This off-diagonal part is all in the upper-triangle, so it is nilpotent; the diagonalizable part we call “semisimple”. And what makes this particular decomposition special is that the two parts commute. Indeed, the block-diagonal form means we can carry out the multiplication block-by-block, and in each block one factor is a constant multiple of the identity, which clearly commutes with everything.
More generally, we will have the Jordan-Chevalley decomposition of an endomorphism: any can be written uniquely as the sum , where is semisimple — diagonalizable — and is nilpotent, and where and commute with each other.
Further, we will find that there are polynomials and — each of which with no constant term — such that and . And thus we will find that any endomorphism that commutes with with also commute with both and .
Finally, if is any pair of subspaces such that then the same is true of both and .
We will prove these next time, but let’s see that this is actually true of the Jordan normal form. The first part we’ve covered.
For the second, set aside the assertion about and ; any endomorphism commuting with either multiplies each block by a constant or shuffles similar blocks, and both of these operations commute with both and .
For the last part, we may as well assume that , since otherwise we can just restrict to . If then the Jordan normal form shows us that any complementary subspace to must be spanned by blocks with eigenvalue . In particular, it can only touch the last row of any such block. But none of these rows are in the range of either the diagonal or off-diagonal portions of the matrix.