Are there $A,B, C, D$ vector spaces such that $A \oplus B \cong C \oplus D$, and $A\cong C$, but $B\not\cong D$?
How to find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is not isomorphic to $D?$