# Nilpotent Lie Algebras: Irreducible Representations

Ok, I'm in contradiction with my book while doing research on the topic of irreducible representations of nilpotent Lie Algebras. I was thinking of proving they are all one-dimensional by Engel's Lemma, but the book says this is only true if the field is closed algebraically. I couldn't find an example over the real numbers that was 2 dimensional or more. Here is my proof:

A representation is irreducible if the only subspaces that are invariant by it are $${0}$$ and the whole vector space. But by Engel's Lemma, since the algebra is nilpotent, their representation also is and there is $$0\not= v \in V$$ such that $$\rho(X)v=0$$ $$\forall X \in \mathfrak{g}$$ , therefore , $$\mathbb{F}v$$ is invariant by $$\rho$$, therefore if it is irreducible, then it has to be $$\mathbb{F}v$$.

We cannot use Engel's theorem (which holds in arbitrary characteristic). Consider the following counterexample for the Heisenberg Lie algebra in characteristic $$p>2$$. It is has an irreducible representation of dimension $$p$$, see here. The assumption of Engel's theorem is that any element of a Lie algebra acts by a nilpotent operator. However, even for an abelian Lie algebra, this need not be true for a representation.