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Is it tough? There are only seven fundamental units. https://en.wikipedia.org/wiki/SI_base_unit

Just represent every unit in terms of them then it's good.

One problem with the array you came up is that the units are not orthogonal, since Joules = Newtons * Meters.

Actually, there's more "kinds" of units: radians (versus degrees) and steradians come to mind.

These are dimensionless derived units in SI equivalence (respectively m/m and m2/m2)

Sorry! I never really spent much time in the physical sciences, so I didn't know that. You would obviously not want to pick Joules as a fundamental unit.

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