If $V=U\oplus U'=W\oplus W'$, then is $V=(U\cap W)\oplus (U'+W')$? I am interested in knowing whether the following fact is true or not for a vector space $V$, and subspaces $U,U',W,W'$.

If $V=U\oplus U'=W\oplus W'$, then $V=(U\cap W)\oplus(U'+W')$


Oh wow I guess this is one of those moments where you realise the answer to your own question as soon as you ask it. If $U'$ and $W'$ were orthogonal complements I think it might be true, but they're not so there's more freedom,
A counterexample is $V=\mathbb{R}^2$, $U=\operatorname{span}\{(1,0)\}, W=\operatorname{span}\{(0,1)\}$ and $U'=W'=\operatorname{span}\{(1,1)\}$. Then $U\cap W=\{(0,0)\}$ and $U'+W'=\operatorname{span}\{(1,1)\}$, hence $(U\cap W)+(U'+W')$ is a proper subspace of $V$.
What seems like should be true, is that $(U\cap W)\cap(U'+W')=\{0\}$, or at least I haven't thought of a counterexample to that yet.
