I am trying to understand the non-commutative analysis for nilpotent Lie groups, so I've been reading Corwin's and Greenleaf's book on the representation theory of nilpotent groups and going through examples of the orbit method.

My problem is that I would like to apply the Fourier transform to the Lie algebra generators, and so basically I only need the unitary representations of the Lie algebra. We can simply differentiate the Lie group representations of one parametric subgroups, but as I was trying to go into more difficult examples, the calculations became quiet tedious very fast, because they rely (at least the way it's done in the book) on the Campbell-Hausdorff formula to construct the induced representations.

So my question is: are there any constructions of irreducible unitary representations of nilpotent Lie groups/algebras, that do not use the Campbell-Hausdorff formula or that are more friendly for high-dimensional cases?

Thanks.