Is the following true: for a vector space $V$ and subspaces $\left\{U_i\right\}_{i\in I}$ and $\left\{W_j\right\}_{j\in J}$ if $$V=\bigoplus_{i\in I} U_i = \bigoplus_{j\in J} W_j$$ is it then true that there is a "refinement" $$V=\bigoplus_{i\in I,j\in J} U_i\cap W_j\textrm{?}$$ If not, is it true provided that $U_i\cap W_j\neq \left\{0\right\}$ for all $i\in I$ and $j\in J$?
*My attempts to prove/disprove: It is clear that we don't necessarily have $$U_i=\bigoplus_{j\in J} U_i\cap W_j$$ e.g. if $W_1$ and $W_2$ are the $x$/$y$ axes in $k^2$ and $U_1$ is the diagonal line $x=y$. This makes me doubt the claim. In fact, I just realised that taking $U_2$ as the line $x=-y$ this gives a counterexample to the original suggestion, though we do have $U_i\cap W_j= \left\{0\right\}$ so it doesn't refute the second claim.
For $V=\mathbb{R}^6$ let's take : $$U_1=\{(x,y,z,x,y,z) / x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{R}\}$$ $$U_2=\{(x,y,z,2x,2y,2z) / x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{R}\}$$ $$W_1=\{(x,y,z,x,2y,3z) / x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{R}\}$$ $$W_2=\{(x,y,z,2x,3y,z) / x\in\mathbb{R},y\in\mathbb{R},z\in\mathbb{R}\}$$ It's easy to verify that $V=U_1\oplus U_2$ and $V=W_1\oplus W_2$ but for $1\leq i,j \leq 2$ we have : $$Dim(U_i\cap W_j)=1$$ thus : $$ Dim(\sum_{\substack{i,j\in \{1,2\}}} \!U_i\cap W_j) \leq \sum_{\substack{i,j\in \{1,2\}}} \!Dim(U_i\cap W_j)=4 <6 $$ Hence V cannot be the direct sum of $U_i\cap W_j$