An *abstract Jordan decomposition* of an element of a Lie algebra L is a decomposition
of the form a = a$\_{s}$ + a$\_{n}$, where


(a) ad a$\_{s}$ is a diagonalizable (equivalently semisimple) endomorphism of L.


(b) ad a$\_{n}$ is a nilpotent endomorphism.


(c) [a$\_{s}$, a$\_{n}$] = 0 .


[This note](http://math.mit.edu/classes/18.745/Notes/Lecture_12_Notes.pdf) defines the abstract Jordan decomposition in an arbitrary Lie algebra. Abstract Jordan decomposition in a Lie algebra is unique **when it exists** iff its centre is zero. It seems that the abstract Jordan decomposition maybe not exist even when its centre is zero, who can show me an example?


The same question is at [here](https://math.stackexchange.com/questions/3406659/abstract-jordan-decomposition-maybe-not-exist) with no answer.



Abstract Jordan decomposition maybe not exist
In $\mathfrak g=\left\{\begin{pmatrix}x&x&y\\0&x&z\\0&0&0\end{pmatrix}:x,y,z\in \mathbf R\right\}$, $\mathrm{ad}$-semisimple and $\mathrm{ad}$-nilpotent elements all have $x=0$; so they don’t span. 


