Necessity of condition $[d,n]=0$ in the Jordan decomposition explicit counterexample

Question

I'm trying to find an example to show that the condition $[d,n]=0$ in the Jordan decomposition is necessary. So, I'm looking for a matrix $x$ which can be written as $x = d + n$ with $d$ diagonalisable, $n$ nilpotent and yet this is not the Jordan decomposition of $x$

Answer 1

Example: $$ D = \pmatrix{1&0\\0&2}, \quad N = \pmatrix{0&1\\0&0}. $$