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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 $\mathbb{F}v$.

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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.

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