I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other related operators that are real analytic w.r.t. the parameters); this requires decomposition of the LCT matrix, and long story short, it is necessary to be able to decompose an arbitrary real matrix $B$ into a product $D¹P¹SP²D²$, with $D¹$ and $D²$ being diagonal, $P¹$ and $P²$ possessing at least one real logarithm each, $S$ being symmetric, and all being real.

So far I could not find any combination of known mutliplicative matrix decomposition methods that arrives at this result; I need help.

Note: Here $D¹$ and $D²$ are completely independent, the $¹$ and $²$ having absolutely no meanings, and so on.

UPDATE: After workup, I found out that it is sufficient to prove the following statement: given any two real symmetric matrices $F¹$ and $F²$, there exist real diagonal matrices $D¹$ and $D²$, s.t. the polar decomposition $D¹F¹=P¹U¹$, $P¹$ being positive semidefinite and $U¹$ unitary, and so on, with the product of $U¹$ with the transpose of $U²$ being symmetric.