Follow-up question re: logarithms of matrix-valued functions known not to have any zero eigenvalues [closed]

Question

Given two parametrised "well-behaved" real skew-symmetric matrix-valued functions (of real variable(s)) $K(x)$ and $L(x)$, is it true that there exists another matrix-valued function $M(x)$ s.t. $e^M=e^K*e^L$ for all real $x$?

A sufficient condition, based on my analysis of the BCH-D integral formula, would be that "$[\ln A, B]$", brackets indicating commutators, $A$ being orthogonal (real) with determinant 1, and $B$ being real skew-symmetric, does not depend on the branch of the logarithm as long as said branch is strictly real.

EDIT: Gave my dead answer back to life by breathing rigour in (vide infra).

Answer 1

Answering my own question.

Comes trivially from the BCH-D integral expression; "$\ln(e^Ke^L)-K-L$" being equal to the iterated integral of ${(M(x)-1)y}/{(M(x)(1-y)+y)}*L$, $M(x)=\operatorname{Ad}(e^Ke^{xL})$, on $[0,1]^2$, the integral over $y$ treated as a Cauchy principal value.