  For any non-principal polarisation $D$, we explicitly construct $D$-polarised
abelian variety $A$, such that its dual abelian variety is not (abstractly)
isomorphic to $A$. For $\dim(A)>3$ the construction includes examples with
submaximal Picard number equal to $(\dim(A)-1)^2+1$. Moreover, we show an
example of an abelian variety that is isomorphic to its dual, yet it does not
admit a principal polarisation.
