Apologies for asking a stupid question. Working on the exercise 7.8 from [Introduction to Lie Algebras](http://gen.lib.rus.ec/book/index.php?md5=EA1B3D286C4131781D32E790254DC3F7). It says:

*Let $L$ be the Heisenberg algebra with basis $f$, $g$, $z$ such that $[f, g] = z$
and $z$ is central. Show that $L$ does not have a faithful finite-dimensional irreducible representation.*

First of all, what is wrong with the traditional upper-diagonal representation? Namely,

$$ f = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}
\quad,\quad
g = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}
\quad,\quad
z = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}
\quad. $$

I would also be great to know why these three requirements for the representation (faithful, finite-dimensional, irreducible) are incompatible for the Heisenberg algebra.

Thanks.