Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 4 / 1 / iv

On ${\mathbb{C}}^3$ take the matrix unitsTheir products give $[E_{12},E_{23}]=E_{13}$, while $E_{13}$ commutes with each of the other two. Thus this is a Lie algebra representation. The three images are linearly independent, so its kernel is zero: it is faithful and finite-dimensional. It is reducible, as expected from the classification in part ii; for example, the first coordinate line is invariant.