Characterizations and properties of good eggs in PG(4n-1,q), q odd

Using the explicit description given in [ M. Lavrauw, T. Penttila, On eggs and translation generalised quadrangles, J. Combin. Theory Ser. A. 96 (2) (2001) 303-315] we prove that a good egg of PG(4n-1,q), q odd, is isomorphic to an ovoid of PG(3,qn) seen over GF(q) if and only if there exists a (3n-1)-space skew from the good element and containing at least five egg elements. There are only three classes of known examples of eggs in PG(4n-1,q), which are not elementary. For each of these classes we obtain strong restraints on the intersection of egg elements with (3n-1)-spaces spanned by three different egg elements.