Common plane for three pairwise complementary subspaces

If $V=A_1\oplus A_2=A_2\oplus A_3=A_1\oplus A_3$ with nonzero subspaces, choose $0\ne v\in A_3$ and write $v=a_1+a_2$ using the first direct sum of vector spaces. Both components are nonzero. The plane spanned by them meets $A_1$ and $A_2$ in their component lines and meets $A_3$ in the line through $v$. Its intersection with $A_3$ cannot be the whole plane, since $a_1\notin A_3$.