Problem source

Let $a,b,c,d,p,q,r$ and $s$ be real numbers. By considering the determinant of the matrix product 
\[
\begin{pmatrix}z_{1} & z_{2}\\
-z_{2}^{*} & z_{1}^{*}
\end{pmatrix}\begin{pmatrix}z_{3} & z_{4}\\
-z_{4}^{*} & z_{3}^{*}
\end{pmatrix},
\]
where $z_{1},z_{2},z_{3}$ and $z_{4}$ are suitably chosen complex numbers, find expressions $L_{1},L_{2},L_{3}$ and $L_{4},$ each of which is linear in $a,b,c$ and $d$ and also linear in $p,q,r$ and $s,$ such that 
\[
(a^{2}+b^{2}+c^{2}+d^{2})(p^{2}+q^{2}+r^{2}+s^{2})=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+L_{4}^{2}.
\]

Solution source

Supppose $z_1 = a+ib, z_2 = c+id, z_3 = p+iq, z_4 = r+is$ then:

\begin{align*}
&& \det \left (\begin{pmatrix}z_{1} & z_{2}\\
-z_{2}^{*} & z_{1}^{*}
\end{pmatrix}\begin{pmatrix}z_{3} & z_{4}\\
-z_{4}^{*} & z_{3}^{*}
\end{pmatrix} \right) &= \det \begin{pmatrix}z_{1} & z_{2}\\
-z_{2}^{*} & z_{1}^{*}
\end{pmatrix}\det\begin{pmatrix}z_{3} & z_{4}\\
-z_{4}^{*} & z_{3}^{*}
\end{pmatrix} \\
&& \det \begin{pmatrix}z_{1}z_3-z_2z_4^* & z_1z_4+z_2z_3^*\\
-z_2^*z_3-z_1^*z_4*& -z_2^*z_4+z_{1}^*z_3^*
\end{pmatrix}&= (z_1z_1^*+z_2z_2^*)(z_3z_3^*+z_4z_4^*) \\
&& |z_{1}z_3-z_2z_4^*|^2+|z_1z_4+z_2z_3^*|^2&= (a^2+b^2+c^2+d^2)(p^2+q^2+r^2+s^2) \\
&& L_1^2 + L_2^2+L_3^2+L_4^2 &= \ldots 
\end{align*}