I was going through chapter 9 of Fulton and Harris, as I am teaching the course, and ran across the example of the Lie algebra $\mathbb{C}$, with the following representation:
\begin{equation} \rho: t \mapsto \begin{pmatrix} t & t \\ 0 & 0 \end{pmatrix} \end{equation}
The book presents this as an example where the matrix is ``neither diagonalizable nor nilpotent'', and where neither the semisimple and nilpotent parts will be in the image of $\rho$. The point they want to showcase is how the semisimplicity of the Lie algebra is crucial.
I'm having trouble with this example, as the matrix in question is already semisimple as it stands.
Can anyone `fix' this example, or show me another neat representation of $\mathbb{C}$ where J.C. decomposition is not preserved?
Sure, just take the representation $$t \mapsto \left( \begin{matrix} t & t \\ 0 & t \end{matrix} \right)=\left( \begin{matrix} t & 0 \\ 0 & t \end{matrix} \right)+\left( \begin{matrix} 0 & t \\ 0 & 0 \end{matrix} \right).$$