<s>[INST] determinant of 4x4-matrix occuring in Zarhin's trick What's the easiest/fastest way of calculating the determinant of $$\begin{pmatrix}a & b & c & d\\ -b & a & -d & c\\ -c & d & a & -b \\ -d & -c & b & a\end{pmatrix}$$? The result is $(a^2+b^2+c^2+d^2)^2$. This determinant occurs in Zarhin's trick (if $A$ is an Abelian variety, $(A \times A^t)^4$ is principally polarised).
 [/INST] Write $X$ for the matrix in question. Then 
$$
X \cdot X^T = (a^2+b^2+c^2+d^2) I. 
$$
This implies that 
$$
\det(X) =f(a,b,c,d)= \pm (a^2+b^2+c^2+d^2)^2 . 
$$
Now as 
$$
f(a,0,0,0)=a^4
$$
for all $a \in \mathbf R,$ we have that
$$
\det(X)=(a^2+b^2+c^2+d^2)^2
$$
for all $a,b,c,d \in \mathbf R.$ 
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