Is there anything in between? In some sense, this is the same example with the reciprocal of the rank.

projective indecomposable modules

Do you know one where both are finite dimensional?

These are not arbitrary nilpotent pairs. They commute.

You are right. I'm not sure what this comment was about...

The argument above doesn't have a restriction on n.

I don't understand this comment at all.

Michal-It is fine for the constants to be the same, but there is no reason why this has to be the case.

The problem is that you might start with a simple Lie algebra over $\mathbb{R}$ which is no longer simple after extending scalars to $\mathbb{C}$ (e.g. $\mathfrac{so}(3,1)$). So the forms are not necessarily proportional over $\mathbb{C}$ and therefore, by your argument, they are not necessarily proportional over $\mathbb{R}$.

I don't think this is a characteristic $p$ issue. The point is that I assummed we were working over an algebraically closed field. Schur's lemma doesn't work over the reals.

You are right, I was assuming the base field was $\mathbb{C}$, and had Schur's lemma in mind.