Proof Since is algebraically closed, then the characteristic polynomial of is of the form: where are the distinct eigenvalues of . Denote the generalised eigenspaces of by . Let . For we have that and are coprime, then by the Chinese Remainder Theorem,
We also require that , noting that this is redundant if is an eigenvalue.
Define . So . Define and . We claim that these have the desired properties. Indeed, , becauseFurthermore, and are polynomials in , and therefore commute with each other and with anything that commutes with . If with , then the same hold by replacing with a polynomial in , in particular this is true for and .
We need to show that is semisimple and is nilpotent. Recall that when acts on , it splits into its generalised eigenspaces as above with , preserving each as . Therefore: and thus acts as a scalar on each of the eigenspaces . Therefore, is semisimple.
On each space has generalised eigenvalue and has eigenvalue as above; hence, on every has generalised eigenvalue 0 . Therefore it is nilpotent.
Finally, we must also show that and are unique. Suppose that for some semisimple and nilpotent with . Then, since , we have:The LHS is a sum of commuting semisimple operators, because they commute with and therefore commute with any operator that commute with , so is semisimple. By the same reasoning the RHS is a sum of commuting nilpotent operators, so is nilpotent. It follows that is semisimple and nilpotent: so it is diagonalisable, but with all eigenvalues equal to 0 ; hence it is the zero operator. That is, , so and .
Proposition
If is semisimple so is , and similarly, if is nilpotent so is .
Furthermore, if is a Jordan-Chevalley decomposition in then is a Jordan-Chevalley decomposition in .
Proof Since is semisimple, has an eigenbasis with eigenvalues . A basis for is given by . Then so diagonalizes in this basis, hence is semisimple. is nilpotent by the lemma. Since is a Lie algebra homomorphism, the polynomial relations are preserved. Finally, the Jordan-Chevalley decomposition is unique, so is the Jordan-Chevalley decomposition of .
The above decomposition can be pulled back to any semisimple Lie algebras through the adjoint representation:
Abstract Jordan-Chevalley Decomposition
Let be a semisimple Lie algebra over an algebraically closed field. For let be the Jordan decomposition of in . Then, there exist unique such that and . The decomposition in is called the abstract Jordan-Chevalley decomposition.
Theorem
For semisimple Lie algebras, the Jordan–Chevalley decomposition is preserved by all finite-dimensional representations. That is, if is semisimple and is ad-semisimple, then is semisimple in any finite dimensional representation of .