I reformulate and detail just in case : $\displaystyle (a,b,c,d)=(1,1,0,1)=\alpha$ and $\displaystyle (e,f,g,h)=(0,-1,1,1)=\beta$.

$\displaystyle (i,j,k,l)=(2,1,1,3)=\gamma$. I noticed that $\displaystyle \gamma = \alpha + 2 \beta$.

I must find a vector $\displaystyle \zeta$ such that $\displaystyle span \{ \gamma, \zeta \} = span \{ \alpha, \beta \}$.

My main question is : say I found a vector $\displaystyle \zeta$ and I want to check if it does span $\displaystyle V$ along with $\displaystyle \gamma$.

Is it enough to check that both $\displaystyle \alpha$ and $\displaystyle \beta$ are linear combination of $\displaystyle \gamma$ and $\displaystyle \zeta$? My intuition says yes because if $\displaystyle \alpha$ and $\displaystyle \beta$ can be written as a comb. linear of $\displaystyle \gamma$ and $\displaystyle \zeta$, so does any vector spanned by $\displaystyle \alpha$ and $\displaystyle \beta$. And I also must check that $\displaystyle \zeta$ is a comb. linear of $\displaystyle \alpha$ and $\displaystyle \beta$. If it is then I can conclude that $\displaystyle span \{ \gamma, \zeta \} = span \{ \alpha, \beta \}$.

Is there a way to find $\displaystyle \zeta$, other than having a mathematical eye?