Before revision:  It relies on a matrix flow decomposition method that finds the coarsest block diagonal flow representation for the hermitean matrix flow H\!K  (t) = \cos(t) H + \sin(t) K  . Here H\!K  (t) is a 1-parameter varying linear combination of the real and skew part matrices H = (A+A^*)/2 and K = (A-A^*)/(2i) of A.
After revision:  It relies on a matrix flow decomposition method that finds the coarsest block diagonal flow representation for the hermitean matrix flow H\!K (t) = \cos(t) H + \sin(t) K if this is possible . Here H\!K (t) is a 1-parameter varying linear combination of the real and skew part matrices H = (A+A^*)/2 and K = (A-A^*)/(2i) of A.