Complements of sum of vector space

Question

Let $V_1,V_2,W_1,W_2,X$ be vector spaces over the field $k$.
Assume that $V_1\oplus W_1=X$ and $V_2\oplus W_2=X$.

Is $(V_1+V_2)\oplus(W_1\cap W_2)=X$ ?

Here $\,V\oplus W=X\,$ means $\,V+W=X\,$ and $\,V\cap W=\{0\}\,$.

Answer 1

No, this is not automatic.

Here's a counter-example in $\mathbb C^2$: Let $\{b_1,b_2\}$ be a basis.
Set $\,V_1=V_2=\langle b_1\rangle$, $W_1=\langle b_1+b_2\rangle$, and $W_2=\langle b_2\rangle\,$.
Then $\,V_1 +V_2 =V_1\,$ and $\,W_1\cap W_2=\{0\}\,$.