I am not sure what you would call this theorem for complex matrices - Jordan-Chevalley decomposition, Jordan normal form or maybe SN decomposition (although here it is really the multiplicative form)!
This is a problem in Lie Groups by Rossmann
Show that any invertible matrix can be uniquely written as $a=bc$ where $b$ is semisimple, $c$ is unipotent and $b$ and $c$ commute
(I"m assuming matrices are complex valued, although it is not stated as such)
The usual proof I know involves using a basis of generalised eigenvectors for $\mathbb{C}^n$, and is reasonably complex, at least for an exercise. (For example, see Appendix B of Lie groups, Lie algebras, and representations: an elementary introduction by Brian Hall)
This leads me to believe there may be a simpler proof, although it seems to be eluding me.
Any hints?