I learned the material in this post from the book by Humphreys on Lie algebras and representation theory.
Recall that if is any algebra (not necessarily associative), then the derivations of form a Lie algebra , and that if is actually a Lie algebra, then there is a homomorphism . In this case, the image of is said to consist of inner derivations.
Theorem 1 Any derivation of a semisimple Lie algebra is inner.
To see this, consider ; by semisimplicity this is an injection. Let the image be , the inner derivations. Next, I claim that . Indeed, if and , we have
In other words, . This proves the claim.
Consider the Killing form on and the Killing form on . The above claim and the definition as a trace shows that .
Now, if is the orthogonal complement to under the form , we must have
but also, because of nondegeneracy of ,
I claim now . If , then for any ,
because is an ideal (by invariance of the Killing form again). By semisimplicity, , and since was arbitrary, we find is the zero derivation. So .
Abstract Jordan decomposition
Now assume our fields are algebraically closed.
Proposition 2 Let be a finite-dimensional algebra. Let be a derivation, and regard as an element of to take its Jordan decomposition . Then are derivations of as well.
It is clearly enough to prove is a derivation. There may be confusion caused by my using to refer to a specific
The idea is to write as a direct sum
where the are -invariant subspaces with acting nilpotently on them. Then acts on by .
It is enough to check that for any ,
Now indeed, we have a binomial-like formula
that can be checked by induction on . It shows that is annihilated by some high power of .
Theorem 3 (Abstract Jordan decomposition) Let be a semisimple Lie algebra. Then we can write any uniquely as where is semisimple, is nilpotent, and commute with each other and with every element of that commutes with .
We imbed as a subalgebra of via the mapping. Then splits into semisimple and nilpotent parts, . Then are derivations of by the proposition, and inner ones by Theorem 1, so we get
and semisimplicity implies then . Since
we find . The commuting properties are thus seen to follow from the corresponding ones of the normal Jordan decomposition. Uniqueness follows by the uniqueness of the Jordan decomposition in .
Note that if we have decompositions , and commute, then
is the Jordan decomposition for .
This by itself is not all that interesting. But it turns out to be the case that semisimple elements (i.e., those whose nilpotent part is zero) in a Lie algebra act by semisimple endomorphisms in any representation. We need to talk about complete reducibility first though.