I am looking for examples of Artinian Gorenstein subalgebras with the same socle degrees. More precisely, let $A$ be an Artinian Gorenstein $k$-algebra (with $k$ algebraically closed of characteristic $0$) with standard grading (generated in degree $1$) and socle degree, say $d$. Let $B \subset A$ be an Artinian Gorenstein subalgebra of $A$. Is it possible that $B$ has the same socle degree $d$?

Similarly, if $B$ is a quotient of $A$ and $B$ is an Artinian, Gorenstein $k$-algera. Is it possible that $B$ has the same socle degree $d$?