What can you tell about the span of the following union set? Consider $\mathbb{C}^5$ and the following three sets, each containing two orthogonal vectors:
$A=\{a_1,a_2 \}$, $B=\{b_1,b_2 \}$ and $C=\{c_1,c_2 \}$.
All these vectors are different and non-zero. Additionally, vectors from different sets are not orthogonal. Clearly, each set spans a two-dimensional subspace of $\mathbb{C}^5$: $\text{dim}(\mathcal{S}_i)=\text{dim}(\text{span}(i))=2$ where $i\in\{A,B,C\}$.
I have the following property: the union of each two sets forms a set of linearly independent vectors, i.e. $\text{dim}(\mathcal{S}_{i\cup j})=4$ for $i \neq j$.
I would like to show that it is not possible that $\text{dim}(\mathcal{S}_{A\cup B \cup C} )=4$.
Does this follow directly from the above (and I'm missing something obvious) or do I need to exploit particular properties of the vectors?

Consider $$A=\{(1,0,0,0,0), (0,1,0,0,0)\}$$
$$B=\{(0,0,1,0,0),(0,0,0,1,0)\}$$
$$C=\{(1,0,1,0,0),(0,1,0,1,0)\}$$
