The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three

Abstract: Let $\pmb k$ be an arbitrary field and $A$ be a standard graded Artinian Gorenstein $\pmb k$-algebra of embedding dimension four and socle degree three. Then, except for exactly one exception, $A$ has the weak Lefschetz property. Furthermore, the exception occurs only in characteristic two.