Decomposition of a 4D double/isoclinic rotation into two simple rotationс

Question

А 4D matrix ($C$) of a double or isoclinic rotation is given. How can it be decomposed into two simple rotations expressed as 4D matrices - $A$ and $B$.

These two rotations need to be along invariant planes therefore $AB = BA = C$.

The problem can be simplified to finding a 4D simple rotational matrix $D$ such that $CD=DC$.

This is because $D=A^{-1}$ or $D=B^{-1}$. WLG let $D=A^{-1}$.

$CD=(BA)A^{-1}=B(AA^{-1})=B$

and

$DC=A^{-1}(AB)=(A^{-1}A)B=B$

Since D is a (proper) simple rotation we know that its four eigenvalues are: 1, 1 and two complex conjugates that lie on the unit circle. And we also know that $det(D)=1$.

I am not sure how to proceed from here.

Answer 1

See "Isoclinic Decomposition" within this Wikipedia article. The key reference is to Van Elfrinkhof (1897).