The sum of joins and sums of decomposition pairs of a vector space

Question

Let $V=S_1\oplus T_1=S_2\oplus T_2$ be a vector space, where $S_1, S_2, T_1, T_2\subseteq V$ are subspaces of $V$ and $\oplus$ denotes the inner direct product. Is it true that $(S_1\cap S_2)+(T_1+T_2)=V$? I think it is true and the sum is direct when $\dim(V) < \infty$, however I failed to construct a counter-example or prove it when $\dim(V)=\infty$.

Answer 1

No, it's not necessarily true. Let $V=\mathbb{R}^2$ with the standard basis $\{e_1,e_2\}$. Let $S_1=\langle e_1\rangle$, $S_2=\langle e_2\rangle$, and $T_1=T_2=\langle e_1+e_2\rangle$. (Here $\langle\cdots\rangle$ means the span.) Then $V=S_1\oplus T_1=S_2\oplus T_2$, as required. But $S_1\cap S_2=\{0\}$ and $T_1+T_2=\langle e_1+e_2\rangle$, so $(S_1\cap S_2)+(T_1+T_2)\neq V$.