Now that we’ve given the proof, we want to mention a few uses of the Jordan-Chevalley decomposition.
First, we let be any finite-dimensional -algebra — associative, Lie, whatever — and remember that contains the Lie algebra of derivations . I say that if then so are its semisimple part and its nilpotent part ; it’s enough to show that is.
Just like we decomposed in the proof of the Jordan-Chevalley decomposition, we can break down into the eigenspaces of — or, equivalently, of . But this time we will index them by the eigenvalue: consists of those such that for sufficiently large .
Now we have the identity:
which is easily verified. If a sufficiently large power of applied to and a sufficiently large power of applied to are both zero, then for sufficiently large one or the other factor in each term will be zero, and so the entire sum is zero. Thus we verify that .
If we take and then , and thus . On the other hand,
And thus satisfies the derivation property
so and are both in .
For the other side we note that, just as the adjoint of a nilpotent endomorphism is nilpotent, the adjoint of a semisimple endomorphism is semisimple. Indeed, if is a basis of such that the matrix of is diagonal with eigenvalues , then we let be the standard basis element of , which is isomorphic to using the basis . It’s a straightforward calculation to verify that
and thus is diagonal with respect to this basis.
So now if is the Jordan-Chevalley decomposition of , then is semisimple and is nilpotent. They commute, since
Since is the decomposition of into a semisimple and a nilpotent part which commute with each other, it is the Jordan-Chevalley decomposition of .