  In this paper, we exploit some geometric-differential techniques to prove the
strong Lefschetz property in degree $1$ for a complete intersection standard
Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove
also some strong Lefschetz properties for the same kind of Artinian algebras in
higher codimensions. Moreover, we analyze some loci that come naturally into
the picture of "special" Artinian algebras: for them, we give some geometric
descriptions and show a connection between the non emptiness of the so-called
non-Lefschetz locus in degree $1$ and the "lifting" of a weak Lefschetz
property to an algebra from one of its quotients.
