  In this article, we are going to search for $n\times n$ matrices $A$ and $B$
such that their generalized numerical range $$W_A(B)=\{tr(AU^*BU) \ :\
U^*U=UU^*=I\}$$ is convex. More specifically, we consider $A=\hat{A}\oplus I_k$
and $B=\hat{B}\oplus I_k$ where $\hat{A}$ and $\hat{B}$ are $2\times 2$. If
$W_A(B)=W_{\hat{A}}(\hat{B})$ then it is a convex set.
