Let $A, B, C$ and $D$ be abelian varieties (over $\mathbb{C}$) such that $A \times B \cong C \times D$, and $A \cong C$. From the irreducibility of abelian varieties, we can say that $B$ and $D$ are isogeneous. But do we actually have $B \cong D$?

This is false even for elliptic curves over $\mathbb{C}$. This was proved by T. Shioda in "Some remarks on abelian varieties" J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 1121, http://repository.dl.itc.utokyo.ac.jp/dspace/bitstream/2261/6164/1/jfs240102.pdf. 

