Consider the subvariety $V$ of $\mathbb{C}^{16} \times \mathbb{C}^{16}$ given by \begin{align*} V = \{ (X, Y) \in \mathbb{C}^{16} \times \mathbb{C}^{16}: X = \left( \begin {array}{cccc} 0&0&-\mu_{{3,4}}&\mu_{{1,4}} \\ 0&0&0&0\\ 0&0&-{\frac {{\mu_{{3,4 }}}^{2}}{\mu_{{1,4}}}}&\mu_{{3,4}}\\ 0&0&0&0\end {array} \right) , Y = \left( \begin {array}{cccc} 0&\nu_{{1,2}}&0&\nu_{{1,4}} \\ 0&{\frac { \left( p\mu_{{1,4}}\nu_{{1,2}}-q\mu_{{1,4}}\nu_{{1,2}}-q\mu_{{3,4}}\nu_{{1,4}} \right) \mu_{{3,4}}}{{\mu_{{1,4}}}^{2}p}}&0&{ \frac {p\mu_{{3,4}}\nu_{{1,4}}+q\mu_{{1,4}}\nu_{{1,2}}+q\mu_{{3,4}}\nu_{{1,4}}}{pX _{{1,4}}}}\\ 0&0&0&0\\ 0&0&0&0 \end {array} \right), \mu_{ij}, \nu_{ij} \in \mathbb{C} \}. \end{align*}
I would like to find the defining ideal of $V$, i.e. to find $I$ in $\mathbb{C}[V]=\mathbb{C}[X_{13}, X_{14}, X_{33},X_{34}, Y_{12}, Y_{14}, Y_{22}, Y_{24}]/I$. I found some relations satified by $X_{ij}, Y_{ij}$: \begin{align} & X_{1,3}^2+X_{1,4} X_{3,3}=0 \\ & (p^2+q^2)Y_{1,2} Y_{2,4} -(p+q)^2 Y_{1,4} Y_{2,2}-p q(Y_{1,2}^2+Y_{2,4}^2)=0. \end{align} But maybe there are some more relations. How could I find all the relations? How to do this in Maple or Macaulay2 or some other math software? The ground field $\mathbb{C}$ can also be changed to $\mathbb{Q}$. Thank you very much.