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

Using the explicit description given in \cite{LAVRAUWPENTTILA2001} we prove that a good egg of $\PG(4n-1,q)$, $q$ odd, is isomorphic to an ovoid of $\PG(3,q^n)$ seen over $\GF(q)$ if and only if there exists a $(3n-1)$-space skew from the good element and containing at least 5 egg elements. There are only 3 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.