Let $K$ be a field with $1+1\neq 0$, $V$ be a vector space over $K$ and $\langle \cdot,\cdot \rangle:V\times V\rightarrow K$ an alternating bilinear form.
Use following fact
- $v\in V \setminus V^{\perp}\Rightarrow \exists v'\in V:\langle v,v' \rangle=1\text{ and }V=span(v,v')\oplus span(v,v')^{\perp} $
to find basis $\mathcal B$ of $\mathbb Q^4$ over the field $\mathbb Q$ such that Gram matrix $\begin{pmatrix}0& -3& 1& -3\\ 3&0&-3&1\\ -1&3&0&0\\ 3&-1&0&0 \end{pmatrix}$ of a standard basis has the form $\begin{pmatrix} 0& 1& 0& 0\\ -1&0&0&0\\ 0&0&0&1\\ 0&0&-1&0 \end{pmatrix}$ of basis $\mathcal B$.